Quasi-Stationary Phenomena in Nonlinearly Perturbed Stochastic Systems 9783110208252, 9783110204377

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de Gruyter Expositions in Mathematics 44

Editors V. P. Maslov, Academy of Sciences, Moscow W. D. Neumann, Columbia University, New York R. O. Wells, Jr., International University, Bremen

Quasi-Stationary Phenomena in Nonlinearly Perturbed Stochastic Systems by

Mats Gyllenberg and Dmitrii S. Silvestrov

≥

Walter de Gruyter · Berlin · New York

Mats Gyllenberg Department of Mathematics and Statistics P.O. Box 68 FIN-00014 University of Helsinki Finland E-Mail: [email protected]

Authors Dmitrii S. Silvestrov Division of Applied Mathematics School of Education, Culture and Communication Mälardalen University P.O. Box 883 SE-72123 Västera˚s Sweden E-Mail: [email protected]

Mathematics Subject Classification 2000: Primary 60F05, 60F10, 60K05, 60K15; secondary 60J22, 60J27, 60K20, 60K25, 65C40 Key words: Nonlinear perturbation, quasi-stationary phenomenon, pseudo-stationary phenomenon, stochastic system, renewal equation, asymptotic expansion, ergodic theorem, limit theorem, large deviation, regenerative process, regenerative stopping time, semi-Markov process, Markov chain, absorption time, queueing system, population dynamics, epidemic model, lifetime, risk process, ruin probability, Crame´r-Lundberg approximation, diffusion approximation

앝 Printed on acid-free paper which falls within the guidelines 앪 of the ANSI to ensure permanence and durability.

ISSN 0938-6572 ISBN 978-3-11-020437-7 Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available in the Internet at http://dnb.d-nb.de. 쑔 Copyright 2008 by Walter de Gruyter GmbH & Co. KG, 10785 Berlin, Germany. All rights reserved, including those of translation into foreign languages. No part of this book may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage or retrieval system, without permission in writing from the publisher. Typeset using the authors’ LATEX Files: Kay Dimler, Müncheberg. Printing and binding: Hubert & Co. GmbH & Co. KG, Göttingen. Cover design: Thomas Bonnie, Hamburg.

Preface

Quasi-stationary phenomena in stochastic systems are a subject of intensive studies that started in 60s of 20th century. These phenomena describe the behaviour of stochastic systems with random lifetimes. The core of the quasi-stationary phenomenon is that one can observe something that resembles a stationary behaviour of the system before the lifetime goes to the end. Examples of stochastic systems, in which quasi-stationary phenomena can be observed, are various queuing systems and reliability models, in which the lifetime is usually considered to be the time in which some kind of a fatal failure occurs in the system. Another class of examples of such stochastic systems is supplied by population dynamics or epidemic models. In population dynamics models, the lifetimes are usually the extinction times for the corresponding populations. In epidemic models, the role of the lifetime is played by the time of extinction of the epidemic in the population. Usually the behaviour of a stochastic system can be described in terms of some Markov type stochastic process ."/ .t / and its lifetime defined to be the time ."/ at which the process ."/ .t / hits a special absorption subset of the phase space of this process for the first time. The object of interest is the joint distribution of the process ."/ .t / subject to a condition of non-absorption of the process up to a moment t , i.e., the probabilities Pf."/ .t / 2 A; ."/ > t g. A mathematical description of quasistationary phenomena is connected with studies of the asymptotic behaviour of these probabilities for t ! 1. A typical situation is when the process ."/ .t / and the absorption time ."/ depend on a small parameter "  0 in the sense that some of their local “transition” characteristics depend on the parameter ". The parameter " is involved in the model in such a way that the corresponding local characteristics are continuous at the point " D 0, if regarded as functions of ". These continuity conditions permit to consider the process ."/ .t /, t  0, for " > 0, as a perturbed version of the process .0/ .t /, t  0. For example, one can consider queuing systems with servers that could be in a working state or in the state of reparation. The lifetime is the first time when all the servers in the system are in reparation. Then we can consider a system for which the reparation can be quickly completed, so that the expected time of the reparation plays the role of a small perturbation parameter. In population dynamics models, some local extinction probabilities can also be regarded as small perturbation parameters. In models with perturbations, it is natural to study the asymptotic behaviour of the probabilities Pf."/ .t / 2 A, ."/ > t g when the time t ! 1 and the perturbation

vi

Preface

parameter " ! 0 simultaneously. Here and henceforth in the book symbol " ! 0 is used to show that 0 < " ! 0, i.e., " tends to 0 being positive. The principal novelty of our studies is that we consider the models with nonlinear perturbations. Local transition characteristics that were mentioned above are usually some scalar or vector moment functionals p ."/ of local transition probabilities for the corresponding processes. By a nonlinear perturbation we mean that these characteristics are nonlinear functions of the perturbation parameter " and that the assumptions made imply that the characteristics can be expanded in an asymptotic power series with respect to " up to and including some order k, p ."/ D p .0/ C pŒ1" C    C pŒk"k C o."k /:

(P.1)

The case k D 1 corresponds to models with usual linear perturbations while the cases k > 1 correspond to models with nonlinear perturbations. It turns out that the relation between the velocities with which " tends to zero and the time t tends to infinity has a delicate influence on the quasi-stationary asymptotics. Without loss of generality it can be assumed that the time t D t ."/ is a function of the parameter ". The balance between the rate of the perturbation and the rate of growth of time can be characterized by the following condition which is assumed to hold for some 1  r  k: "r t ."/ ! r < 1 as " ! 0:

(P.2)

The main result of our studies is the so-called mixed ergodic and limit/large deviation theorems given in a form of exponential expansions for the probabilities Pf."/ .t ."/ / 2 A; ."/ > t ."/ g. Under the mentioned above assumptions on the perturbations and some additional natural conditions of Cram´er type, we obtain exponential asymptotic relations of the following form: Pf."/ .t ."/ / 2 A; ."/ > t ."/ g ! Q .0/ .A/e r ar as " ! 0: expf..0/ C a1 " C    C ar 1 "r 1 /t ."/ g

(P.3)

With these relations, we give explicit algorithms for calculating the coefficients .0/ and a1 ; : : : ; ak as functions of the coefficients in the expansions of the local transition characteristics that appear in the initial nonlinear perturbation conditions of type (P.1). It is a non-trivial problem due to the nonlinear character of the perturbations involved. The asymptotic behaviour of Pf."/ .t / 2 A; ."/ > t g is very much different in the two alternative cases: (a) .0/ D 0 and (b) .0/ > 0. In the first case, the absorption times ."/ are stochastically unbounded random variables, that is, they tend to 1 in ."/ probability as " ! 0. In the second case, the absorption times 0 are stochastically bounded as " ! 0. In the literature, the asymptotics related to the second model is known as quasistationary asymptotics. To distinguish between the asymptotics (P.3) in the first and the second cases, we coin the term pseudo-stationary for the first one.

Preface

vii

Another class of asymptotic expansions systematically studied in the book concerns the so-called quasi-stationary distributions. Under some natural moment, communication and aperiodicity conditions imposed on the local transition characteristics there exists a so-called quasi-stationary distribution for the process ."/ .t /, which is given by the formula  ."/ .A/ D lim Pf."/ .t / 2 A j ."/ > t g: t!1

(P.4)

Using the asymptotics (P.3) we also obtain asymptotic expansions for the quasistationary distributions of the nonlinearly perturbed processes ."/ .t /,  ."/ .A/ D  .0/ .A/ C g1 .A/" C    C gk .A/"k C o."k /:

(P.5)

As in the case of asymptotics (P.3), we give an explicit recurrence algorithm for calculating the coefficients g1 .A/; : : : ; gk .A/ as functions of the coefficients in the expansions for local transition characteristics of the processes ."/ .t / in the corresponding initial perturbation conditions. The classes of processes for which this program is realised include nonlinearly perturbed regenerative processes, semi-Markov processes, and continuous time Markov chains with absorption. Our approach is based on advanced techniques, developed in the book, of nonlinearly perturbed renewal equations. Applications to the analysis of quasi-stationary phenomena in models of nonlinearly perturbed stochastic systems considered in the book pertain to models of highly reliable queueing systems, M/G queueing systems with quick service, stochastic systems of birth-death type, including epidemic and population dynamics models, metapopulation dynamic models, and perturbed risk processes. As was mentioned above, quasi-stationary phenomena were a subject of intensive studies during several decades. An extensive survey of the literature and comments can be found in the bibliographical remarks given in the book. The results related to the asymptotics given in (P.3) and (P.5) and known in the literature mainly cover the case k D 1, which corresponds to the model of linearly perturbed processes. Some known results also relate to the case where the perturbation condition (P.1) has the special form of relation (c) p ."/ D p .0/ C pŒ1". In context of nonlinearly perturbed models, this is a particular case of nonlinear perturbation condition (P.1) which holds for every k  1 with all higher order terms pŒ2; pŒ3; : : :  0. It should be noted that the asymptotic expansions given in (P.3) and (P.5) obviously cover this case but not vise versa. Indeed, the asymptotic expansions obtained for the models with perturbation condition of the form (c) do not give any information about contribution of the terms pŒ2; pŒ3; : : : in nonlinear asymptotic expansions given in (P.3) and (P.5) for the models where these higher order terms take non-zero values. Throughout the book, we shall make use of several basic types of conditions labeled by letters that make them suggestive. These classes of conditions are the following:

viii

Preface

A for Absorption conditions; B for Balancing conditions which connect and balance the rate of time growth with the rate of the perturbation; C for Cram´er type conditions, that is, the conditions concerning the asymptotic finiteness of the exponential moments of the distribution functions that generate the corresponding renewal equations; D is for weak convergence conditions on the Distribution functions generating the corresponding renewal equations; E for conditions of Ergodicity for the corresponding stochastic processes; F for continuity conditions on the Forcing function of the corresponding renewal equations; P for Perturbation conditions, that is, the conditions relating the model ingredients with the perturbation parameter " in terms of expansions of local transition characteristics in asymptotic power series in the parameter ", etc. Conditions belonging to a specific class have subscripts numbering conditions in the class. Local conditions used in theorems, lemmas, definitions or remarks are indicated with small Greek letters in parentheses as .˛/, .ˇ/, etc. Local conditions in the text can be indicated as (a), (b), etc. The indicated conditions are referred to always within the subsection where these conditions are introduced. Subsections, theorems, lemmas, definitions and remarks have a triple numeration. For example, Theorem 1.2.3 means Theorem 3 of Section 1.2. Formulas also have a triple numeration. For example, label (1.2.3) refers to Formula 3 in Section 1.2. We finally comment on the structure of the book. The book contains an extended introduction, where we present in informal form the main problems, methods, and algorithms that constitute the content of the book. In Chapters 1 and 2 we present results which deal with a generalisation of the classical renewal theorem to a model of the perturbed renewal equation. These results are interesting by their own and, as we think, can find various applications beyond the areas mentioned in the book. In Chapters 3, 4, and 5 we study asymptotics of the types (P.3) and (P.5) for nonlinearly perturbed regenerative processes, semi-Markov processes, and continuous time Markov chains with absorption. Chapters 6 and 7 are devoted to applications of the theoretical results to studies of quasi-stationary phenomena for various nonlinearly perturbed models of stochastic systems. In Chapter 6, quasi-stationary phenomena are studied for highly reliable queueing systems, M/G queueing systems with quick service, stochastic systems of birth-death type, including epidemic and population dynamics models, and metapopulation dynamic models; Chapter 7 deals with perturbed risk processes. The last Chapter 8 contains three supplements. The first one gives some basic operation formulas for scalar and matrix asymptotic expansions. In the second supplement we discuss some new prospective directions for future research in the area. As was mentioned above, the quasi-stationary phenomena and related problems was a subject of intensive studies during several decades. In the last supplement, we give bibliographical remarks to the bibliography that includes more than 1000 references. We hope that publication of this new book and a comprehensive bibliography of works related to asymptotic studies of quasi-stationary phenomena for perturbed stochastic processes and systems will be a useful contribution to the continuing intensive

Preface

ix

studies in this area. In addition to its use for the research and reference purposes, the book can also be used in special courses on the subject and as a complementary reading in general courses on stochastic processes. In this respect, it may be useful for specialists as well as doctoral and advanced undergraduate students. We would also like to thank all our colleagues at the Department of Mathematics and Statistics at the University of Helsinki and the Division of Applied Mathematics of the School of Education, Culture, and Communication at the M¨alardalen University for creating an inspiring research environment and friendly atmosphere which stimulated our work. Helsinki – V¨aster˚as March 2008

Mats Gyllenberg Dmitrii Silvestrov

Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

v

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1 Perturbed renewal equation . . . . . . . . . . . . . . . . . 1.1 Renewal equation . . . . . . . . . . . . . . . . . . . . . 1.2 Renewal theorem for perturbed renewal equation . . . . 1.3 Improper perturbed renewal equation . . . . . . . . . . . 1.4 Perturbed renewal equation under Cram´er type conditions

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21 23 28 48 56

2 Nonlinearly perturbed renewal equation . . . . . . . . . . . . . . 2.1 Exponential expansions for perturbed renewal equation . . . . . 2.2 Exponential expansions for improper perturbed renewal equation 2.3 Asymptotic expansions for renewal limits . . . . . . . . . . . .

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73 74 86 94

3 Nonlinearly perturbed regenerative processes . . . . . . . . . . . . . . 103 3.1 Regenerative processes and regenerative stopping times . . . . . . . . 106 3.2 Mixed ergodic and limit theorems for perturbed regenerative processes 112 3.3 Mixed ergodic and large deviation theorems for perturbed regenerative processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 3.4 Exponential expansions for nonlinearly perturbed regenerative processes146 3.5 Asymptotic expansions for quasi-stationary distributions . . . . . . . 155 4 Perturbed semi-Markov processes . . . . . . . . . . . . . . . . . . . . . 165 4.1 Semi-Markov processes with absorption . . . . . . . . . . . . . . . . 168 4.2 Perturbed semi-Markov processes . . . . . . . . . . . . . . . . . . . 173 4.3 Asymptotics of cyclic absorption probabilities . . . . . . . . . . . . . 192 4.4 Mixed ergodic and limit theorems for perturbed semi-Markov processes 200 4.5 Asymptotic solidarity properties for hitting times . . . . . . . . . . . 216 4.6 Mixed ergodic and large deviation theorems for perturbed semi-Markov processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 5 Nonlinearly perturbed semi-Markov processes . . . . . . . . . . . . . . 257 5.1 Asymptotic expansions for absorption and hitting probabilities . . . . 259 5.2 Asymptotic expansions for power moments of hitting times . . . . . . 280

xii

Contents

5.3 5.4 5.5 5.6

Asymptotic expansions for power-exponential moments of hitting times 289 Exponential expansions for nonlinearly perturbed semi-Markov processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 Asymptotics of quasi-stationary distributions for nonlinearly perturbed semi-Markov processes . . . . . . . . . . . . . . . . . . . . . . . . . 321 Nonlinearly perturbed continuous-time Markov chains . . . . . . . . 350

6 Quasi-stationary phenomena in stochastic systems . . . . . . . . . . . . 6.1 Queueing systems with highly reliable main servers . . . . . . . . . . 6.2 M/G queueing systems with quick service . . . . . . . . . . . . . . . 6.3 Exponential asymptotics for M/G queueing systems with quick service 6.4 Quasi-stationary phenomena in birth-and-death type stochastic systems 6.5 Quasi-stationary phenomena in metapopulation models . . . . . . . .

359 361 372 398 408 426

7 Nonlinearly perturbed risk processes . . . . . . . . . . . . . . . . . . . 7.1 Cram´er–Lundberg and diffusion approximations . . . . . . . . . . . . 7.2 Diffusion approximation for perturbed risk processes . . . . . . . . . 7.3 Asymptotic expansions in Cram´er–Lundberg approximation for perturbed risk processes . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Approximations for ruin probabilities based on asymptotic expansions 7.5 Asymptotic expansions for the distribution of the surplus prior to and at the time of a ruin . . . . . . . . . . . . . . . . . . . . . . . . . . .

439 441 444

8 Supplements . . . . . . . . . . . . . . . . 8.1 Arithmetics of asymptotic expansions 8.2 New directions for the research . . . . 8.3 Bibliographical remarks . . . . . . . .

477 477 482 490

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454 467 473

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 577

Introduction

This book is devoted to studies of quasi-stationary phenomena for nonlinearly perturbed stochastic processes and systems. The introduction intends to present in an informal form the main problems, methods, algorithms, and asymptotic results that constitute the content of the book. We begin from presentation of a simple example which let us outline the circle of problems and illustrate results presented in the book. Then, we present and comment the content of the book by chapters. We try to show the logic and the ideas that make a basis for new methods and algorithms of asymptotic analysis for nonlinearly perturbed stochastic processes and systems developed in the book, to explain the choice and to comment applications which illustrate the theoretical results as well as to comment new directions for future research that could be a natural continuation of the research studies which results are presented in the book.

A model example and the basic quasi-stationary asymptotics Let us consider a M/M queueing system that consists of two servers functioning independently of each other with exponential life and repairing times. The failure and the repairing intensities of the server i are qi; and qi;C , respectively. One can interpret this queueing system as a simple model of a communication system with two channels or a power system with two electric generators. The state of the server i at instant t is given by a random indicator variable i .t / that takes the value 1 if the server i works at this instant, and the value 0 if the server i is repaired at this instant. The state of the system is described by the vector process .t / D .1 .t /; 2 .t //, t  0, which is, in this case, a continuous time homogeneous Markov chain. It has the phase space X D fiN D .i1 ; i2 /; i1 ; i2 D 0; 1g. Figure 1 shows the transition graph, the transition probabilities of discrete time imbedded Markov chain, and the parameters of exponential sojourn times for the process .t /. Let us denote jN the first time at which the process .t / hits into the state jN 2 X. We interpret the hitting time 0N , where 0N D .0; 0/, as a lifetime of the queueing system. By the definition, it is the first time when both servers are out of work. The main object of our interest is the probability PiN f.t / D jN; 0N > t g that the N will be observed in state jN ¤ 0N at time system, which began to function in state iN ¤ 0, t and, the lifetime 0N will not yet end up to time t .

2

Introduction $

'

.1; 1/ q1; C q2; & q2;C q1; C q2;C

'

q2; q1; C q2;

$

.1; 0/



I @ % @@ q1;C q1; @ @ q1; C q2; @ @ q1;C C q2; @@' $ @@ R @ .0; 1/ q1;C C q2;  & %

q1; C q2;C @ I & % @ @ q1; @ @ q1;C q2;C q1; C q2;C @ @ q1;C C q2;C q1;C C q2;C @ @' $ # @@ R @ .0; 0/ q1;C C q2;C " ! & %

q2; q1;C C q2;

Figure 1 The transition graph for the Markov chain .t /. Let us assume that (a) all intensities qi;˙ > 0; i D 1; 2. In this case, it is well known that the probability defined above has the following asymptotics PiN f.t / D jN; 0N > t g e ./ t

N ! QQ iN jN ../ / as t ! 1; iN ; jN ¤ 0:

(0.1)

The exponential normalising coefficient ./ is the unique root of the following nonlinear characteristic equation (b) E1N e 1N .1N  0N / D 1, where 1N D .1; 1/. The limits QQ iNjN ../ /; iN ; jN ¤ 0N are positive numbers which can depend on the initial state iN ¤ 0. They determine probabilities by the formulas Pso-called quasi-stationary ./ ./ ./ Q Q N N Q kN rN . /; j ¤ 0. The quasi-stationary probabili(c) jN . / D Q kN jN . /= r¤ N 0N  ties do not depend on the choice of the state kN ¤ 0N in these formulas. The following asymptotic relation, which is implied, in an obvious way, by relation (0.1), clarifies this term, N PiN f.t / D jN=0N > t g ! jN ../ / as t ! 1; iN ; jN ¤ 0:

(0.2)

The asymptotic relation (0.2) shows that the conditional distribution of the process .t / (conditioned by the event that the lifetime will not end before time t ) stabilises as time t tends to infinity. Thus, one can observe something that resembles a stationary behaviour of the system before the lifetime goes to the end. This is the sense of quasistationary phenomenon. The asymptotic relations (0.1) and (0.2) are used in the same way as usual stationary type asymptotic relations. These relations let one approximate the quantities on the left

3

Introduction

hand side in these relations by the corresponding quantities on the right hand side, for large values of t . As well known, the asymptotic relations of type (0.1) and (0.2) may loose their effectiveness if the parameters of the system take some nearly critical values. Here, it is the case of model with highly reliable servers, where one or both failure intensities q1; and q2; take very small values. ."/ An usual formalism here is to assume that the transition intensities qi;˙ D qi;˙ , ."/

i D 1; 2 depend on a small perturbation parameter "  0 in such way that (d) qi;˙ ! .0/

."/

qi;˙ as " ! 0, i D 1; 2. In this case, the process ."/ .t / and the lifetime 0N also depend on the perturbation parameter ". The parameter " is involved in the model in such a way that the corresponding transition intensities are continuous at the point " D 0, if regarded as functions of ". This continuity condition permits to consider the process ."/ .t /, for " > 0, as a perturbed version of the process .0/ .t /. An intuitive hypothesis is that, in this case, the character of asymptotics should also depend somehow of balance between large time values t and small values of failure ."/ ."/ intensities q1; and q2; . .0/ > 0, i D 1; 2. As It is natural to assume that the limit repairing intensities (e) qi;C .0/

far as the limit failure intensities qi; , i D 1; 2 are concerned, there are three alternatives: (f) both limit failure intensities are positive; (g) one of limit failure intensities is positive while another equal zero; and (h) both limit failure intensities equal zero. The cases (g) and (h) correspond to the model with asymptotically reliable servers. The most well investigated is the case of so-called linearly perturbed systems, where ."/ the pre-limit intensities can be represented in the following asymptotic form (i) qi;˙ D .0/

qi;˙ C qi;˙ Œ1" C o."/ as " ! 0, where jqi;˙ Œ1j < 1, i D 1; 2. .0/

.0/

Let us, for example, assume that (g1 ) q1; D 0 while q2; > 0. This means the the first “main” server is asymptotically absolutely reliable while the second “supplementary” server is not. It turns out that the relation between the velocities with which " tends to zero and the time t tends to infinity has a delicate influence on the quasi-stationary asymptotics. Without loss of generality it can be assumed that the time t D t ."/ is a function of the parameter ". It is well known that in this case and under additional assumption (j) "t ."/ !  as " ! 0, where 0   < 1, the following asymptotic relation takes place, ˚  ."/ .0/ N PiN ."/ .t ."/ / D jN; 0N > t ."/ ! QQ N N .0/ e a1 as " ! 0; iN ; jN ¤ 0: (0.3) ij

Here, coefficient a1 is explicitly determined by the coefficients penetrating the linear perturbation condition (i). In particular, a1 > 0 if q1; Œ1 > 0. As far as .0/ .0/ the coefficients QQ N.0/ QN , N .0/ are concerned, they, as expected, take the form fN  ij .0/ where Q jN are the stationary ˚ .0/ .0/  PiN 1N < 0N .

i

probabilities for the limiting process

.0/ .t /

and

j .0/ fiN D

4

Introduction

It should be noted that there is a principle difference between the asymptotic relations (0.1) and (0.3). In the first one, the lifetime 0N is a finite random variable. The asymptotic relation (0.1) provides information about asymptotic behaviour of the mixed tail probabilities PiN f.t / D jN; 0N > t g, which, according to this relation, can be approximated with the zero asymptotic relative error by the exponential type mixed tail probabilities ./ QQ iNjN ../ / e  t as t ! 1. It is a typical quasi-stationary asymptotic relation. are stochastically unbounded random In the second one, the random variables ."/ 0N variables, that is, they tend to 1 in probability as " ! 0. The asymptotic relation (0.3) presents some kind of mixed ergodic (for the process ."/ .t /) and limit (for the lifetime ."/ 0N ) theorem. According to this relation the state of the perturbed process ."/ .t ."/ / at ."/

the moment t ."/ and the normalised lifetime "0N are asymptotically independent and the joint limit distribution is the product of the stationary distribution fN.0/ Q .0/ N for the i

j

limit process .0/ .t / and the limit exponential distribution e a1 for the normalised lifetimes. The asymptotic relation (0.3) should be classified in some different way. We coin the term “pseudo-stationary” for such kind of asymptotics.

United quasi- and pseudo-stationary asymptotics for nonlinearly perturbed model The question arises if it is possible to get the asymptotic relations given above in the frame of one united approach? The theory presented in the book gives an affirmative answer to this question. The linear perturbation condition (i) does require some special analysis. In many cases, the perturbation parameter " has some natural sense. For example, it can be some kind of “reciprocal cost” or “load rate” parameter such that failure intensities decrease if parameter " tends to zero. In such cases transition intensities may actually be nonlinear functions of this parameter. The linear perturbation condition (i) appears as the result of well-known linearization procedure. The question arise, what effect has this linearization on the quality of the corresponding asymptotic formulas? In this context, it is natural to introduce a nonlinear perturbation condition accord."/ ing to which the transition intensities qi;˙ can be expanded in an asymptotic power series with respect to " up to and including some order k D 0; 1; : : :: ."/

.0/

P.k/ : qi;˙ D qi;˙ C qi;˙ Œ1" C    C qi;˙ Œk"k C o."k /, where jqi;˙ Œnj < 1 for n D 1; : : : ; k and i D 1; 2. It is useful to note that these conditions have a natural subordination in the sense that, for every k D 0; 1; : : :, condition P.k/ is weaker than condition P.kC1/ . In the case k D 0, condition P.k/ reduces to the weak perturbation condition (d), while, in the case k D 1, it coincides with the linear perturbation condition (i).

Introduction

5

The methods developed in the book let us improve the asymptotics presented in relations (0.1) and (0.3) for models with nonlinearly perturbed parameters. In particular, we answer the question in what way the higher order terms in the perturbation condition P.k/ influence the corresponding asymptotic relations. The key role in the asymptotic methods developed in this book is played by the asymptotic expansion for the unique root ."/ of the perturbed characteristic equation, ."/

E1N e 1N .."/  ."/ /D 1N 0N

Z

1 0

˚  D 1: e u P1N ."/ 2 du; ."/  ."/ 1N 1N 0N

(0.4)

We begin our analysis from the case k D 0, where the perturbation condition P.0/ is assumed. This condition implies the asymptotic relation (k) ."/ ! .0/ as " ! 0. Using the methods developed in the book and relation (k) we can write down the asymptotic relation that generalises relation (0.1) to the case where the weak perturbation condition (d) holds. The following asymptotic relation takes place for any 0  t ."/ ! 1 as " ! 0, ."/

PiN f."/ .t ."/ / D jN; 0N > t ."/ g ."/ ."/ e  t

.0/ N N jN ¤ 0: ! QQ N N ..0/ / as " ! 0; i; ij

(0.5)

Also, conditions (d), (e) and (f) imply that for every " small enough all intensities ."/ qi;˙ > 0, i D 1; 2. Thus, the asymptotic relations (0.1) and (0.2) can be written down for the perturbed process ."/ .t /, and the corresponding quasi-stationary distribution ."/  N .."/ /, jN ¤ 0, can be defined by formulas (c) applied to this perturbed process. j Relation (0.5) is also supplemented by the asymptotic relation ."/ .0/ N  N .."/ / !  N ..0/ / as " ! 0; jN ¤ 0: j

j

(0.6)

The asymptotic behaviour of probabilities PiN f."/ .t / D jN; ."/ > t g is very much 0N .0/ .0/ different in the two alternative cases: (1)  D 0 and (2)  > 0. The case (2) .0/ > 0 corresponds to the model where conditions (d), (e), and (f) ."/ hold. In this case, the lifetimes 0N are stochastically bounded as " ! 0. The asymptotic relations (0.5) replaces the asymptotic relation (0.1) for the case of model with perturbed parameters. In the literature, the asymptotics similar with (0.1) is referred as quasi-stationary asymptotics. The case (1) .0/ D 0 corresponds to the model where conditions (d), (e), and are stochastically unbounded random (g) or (h) hold. In this case, the lifetimes ."/ 0N variables, that is, they tend to 1 in probability as " ! 0. To distinguish between the asymptotics (0.5) in these two cases, we, as was mentioned above, use the term “pseudo-stationary” in the case (1).

6

Introduction

Expansions in quasi- and pseudo-stationary asymptotics for nonlinearly perturbed model The asymptotic relation (0.5) looks nicely but has actually a serious drawback. The exponential normalisation with the coefficient ."/ is not so effective because of this coefficient is given us only as the root of the nonlinear equation (0.4), for every "  0. In order to improve the asymptotics provided by relations (0.5) and (0.6), we should assume that the perturbation condition P.k/ , which is stronger than P.0/ , holds for some k  1. In this case, we show that the following asymptotic expansion holds for the characteristic root ."/ , ."/ D .0/ C a1 " C    C ak "k C o."k /:

(0.7)

We also give the explicit effective algorithm for calculating the coefficients a1 ; : : : ; ak as functions of the coefficients in the expansions involved in the nonlinear perturbation condition P.k/ . It is a non-trivial problem due to the nonlinear character of the perturbations. Now, we can involve in the analysis balancing conditions for the perturbation parameter " and time t ."/ . These conditions describe asymptotical time zones of different order. We can assume that the following balancing condition holds for some 1  r  k: B.r/ : "r t ."/ ! r as " ! 0, where 0  r < 1. Using the perturbation condition P.k/ and the balancing condition condition B.r/ we can write down the following asymptotic relation that covers in the united form both cases of pseudo- and quasi-stationary asymptotics, ."/ PiN f."/ .t ."/ / D jN; 0N > t ."/ g .0/ r1 ."/ e . Ca1 "CCar1 " /t

.0/ N ! QQ N N ..0/ /e ar r as " ! 0; iN ; jN ¤ 0: ij

(0.8)

This asymptotic relation is also supplemented by the asymptotic expansions for the quasi-stationary probabilities, ."/ .0/ N  N .."/ / D  N ..0/ / C gjN Œ1" C    C gjN Œk"k C o."k /; jN ¤ 0: j

j

(0.9)

We also give the explicit effective algorithm for calculating the coefficients gjN Œ1; : : : ; gjN Œk as functions of the coefficients in the expansions involved the nonlinear perturbation condition P.k/ . This algorithm essentially use the asymptotic expansion (0.7) but also does require several additional non-trivial steps. Relations (0.8) and (0.9) essentially improve both asymptotic relations (0.5) and (0.6) replacing these simple convergence relations by the corresponding asymptotic expansions. The exponential normalisation with the coefficient .0/ C a1 " C    C ar1 "r1 involves the root .0/ . To find it one should solve only one nonlinear equation (0.4),

7

Introduction

for the case " D 0. As far as the coefficients a1 ; : : : ; ar1 are concerned, they are given in the explicit algebraic recurrence form. Moreover, the root .0/ D 0 in the most interesting pseudo-stationary case. Here, the non-linear step connected with finding of the root of equation (0.4) can be omitted. Relations (0.8) let us get a new type of so-called mixed ergodic (for the process ) theorems. ."/ .t /) and large deviation (for the lifetime ."/ 0N

Let us, for example, consider the pseudo-stationary case, where (1) .0/ D 0 and conditions (d), (e), and (g1 ) hold. If k D 1, then the only case r D 1 is possible for the balancing condition. The asymptotic relation (0.8) reduces to the asymptotic relation (0.3), i.e., it yields the corresponding mixed ergodic and limit theorem for linearly perturbed system. If k D 2, then two cases, r D 1 and r D 2, are possible for the balancing condition. In the case r D 1, the asymptotic relation (0.8) reduces to (0.3). It show that the > "t ."/ g can be approximated by the mixed tail probabilities PiN f."/ .t ."/ / D jN; "."/ 0N .0/

.0/

."/

exponential type mixed tail probabilities fN Q N e a1 "t as " ! 0, with the zero i j asymptotic relative error, in every asymptotic time zone which is determined by the relation "t ."/ ! 1 as " ! 0, where 0  1 < 1. In the case r D 2, the asymptotic relation (0.8) shows that the mixed tail proba."/ bilities PiN f."/ .t ."/ / D jN; "0N > "t ."/ g can be approximated by the the exponential a1 "t type mixed tail probabilities fN.0/ Q .0/ N e a2 2

i

."/

j

as " ! 0, with the asymptotic relative

error 1  e , in every asymptotic time zone which is determined by the relation 2 ."/ " t ! 2 as " ! 0, where 0  2 < 1. If 2 D 0, then "t ."/ D o."1 /, and the asymptotic relative error is 0. Note that this case also covers the situation where "t ."/ is bounded, which corresponds to the asymptotic relation (0.8) with r D 1. This is already an extension of this asymptotic relation since it is possible that "t ."/ ! 1. If 2 > 0, then "t ."/ D O."1 /, and the asymptotic relative error is 1  e 2 a2 . It differs from 0. Therefore, o."1 / is an asymptotic bound for the large deviation zone with zero asymptotic relative error in the above approximation. To get the approximation with zero asymptotic relative error in the asymptotic time zone which are determined by the relation "2 t ."/ ! 2 one should approximate the > "t ."/ g by the exponential type mixed tail probabilities PiN f."/ .t ."/ / D jN; "."/ 0N

mixed tail probabilities fN.0/ Q .0/ e .a1 "Ca2 " /t  fN.0/ Q .0/ e a1 "t a2 2 , i.e., to i jN i jN introduce the corresponding corrections for the parameters in the exponents. Relation (0.8) can be interpreted as a new type mixed ergodic and large deviation ."/ theorem for the nonlinearly perturbed process ."/ .t / and the lifetime 0N . A similar interpretation can be made for the asymptotic relation (0.8) if k > 2. As above, relation (0.8) can also be interpreted as a mixed ergodic and limit theorem if r D 1. On the other hand, this relation is naturally to consider as a mixed ergodic and 2

."/

."/

8

Introduction

large deviation theorem for the corresponding asymptotic time large deviation zones if 1 < r  k. A similar interpretation remains also in the quasi-stationary case, where (2) .0/ > 0. It is also worth to note that some additional problems should be solved to complete the analysis. First of all, these are solidarity propositions which clarify the role of initial state ."/ .0/ in the corresponding asymptotic relations. The question about pivotal properties of the asymptotic expansions (0.7), i.e., what is the first non-zero coefficient in this expansion, is also of a great interest. In conclusion, we would like to mention that the full analysis of perturbed queueing systems similar with those investigated above, but with m servers, is given in Section 6.1 of the book.

Contents of the book The book is devoted to studies of quasi- and pseudo-stationary phenomena for nonlinearly perturbed stochastic processes and systems. The methods used are based on the exponential asymptotics for the nonlinearly perturbed renewal equation. These methods let us get so called mixed ergodic and large deviation theorems and asymptotic expansions in these theorems for nonlinearly perturbed regenerative processes, and then for nonlinearly perturbed semi-Markov processes and Markov chains. Finally, these results are applied to nonlinearly perturbed queueing systems, population dynamics, and epidemic models as well as nonlinearly perturbed risk processes. The book also contains an extended bibliography of works in the area. Let us present in more details the content of the book by chapters.

Chapter 1 In this chapter, we present the theory that can be characterised as a generalisation of classical renewal theory to the model of perturbed renewal equation. The object of our studies is the classical renewal equation Z t ."/ ."/ x ."/ .t  s/F ."/ .ds/; t  0: (0.10) x .t / D q .t / C 0

The central role in the renewal theory is played by the renewal theorem. This theorem gives conditions for existence of a limit x ."/ .1/ for the solution of the renewal equation x ."/ .t / as t ! 1. The importance of the renewal equation and the renewal theorem is based on the fact that distributions of regenerative processes, Markov processes, semi-Markov processes, etc., satisfy renewal type equations. This makes the approximation and asymptotic results connected with the renewal equation a very effective tool for getting ergodic and limit theorems for the processes mentioned above. In Chapter 1, we give a natural generalisation of the classical renewal theorem to the model of perturbed renewal equation, where the forcing function q ."/ .t / and the

Introduction

9

distribution function F ."/ .s/ generating this equation depend on a perturbation parameter "  0 and converge, in some natural sense, to q .0/ .t / and F .0/ .s/ as " ! 0, correspondingly. These continuity conditions allow to consider equation (0.10), for " > 0, as a perturbation of equation (0.10) for " D 0. In the perturbed model, the solution x ."/ .t / of the renewal equation (0.10) also depends on the parameter ", and the problem is to describe the asymptotic behaviour of the solution x ."/ .t / in the so-called triangular array mode, where t ! 1 and " ! 0, simultaneously. Without loss of generality, one can assume that the time t D t ."/ is a function of the perturbation parameter ". It is important to generalise the renewal theorem to the model of the perturbed renewal equation in such a way so that, in the case of the non-perturbed equation, the conditions should reduce to the conditions of the classical renewal theorem mentioned above. These results are presented in Chapter 1. The most interesting asymptotics appears in the case where the distributions F ."/ .s/ may be improper, i.e., F ."/ .1/  1 but for the defect of this distribution, f ."/ D 1  F ."/ .1/, we have f ."/ ! 0 as " ! 0. In this case, the condition balancing the rate at which the time t D t ."/ approaches infinity and the convergence rate of f ."/ to zero as " ! 0 has a delicate influence upon the asymptotics of the solution x ."/ .t /. We study this asymptotics under minimal conditions analogous to those used in the classical renewal theorem but also for the case where the distributions F ."/ .s/ satisfy so-called Cram´er type conditions for exponential moments of these distributions and the limit distribution F .0/ .s/ may be either proper or improper. Asymptotic results presented in Chapter 1 make a basis for obtaining asymptotic exponential expansions for nonlinearly perturbed renewal equations.

Chapter 2 This chapter is devoted to asymptotic exponential expansions for nonlinearly perturbed renewal equations. We impose some Cram´er type condition on the distribution F ."/ .s/ generating the perturbed renewal equation. This condition guarantee existence of the unique root ."/ of the characteristic equation Z 1 e s F ."/ .ds/ D 1: (0.11) 0

Note that the root ."/ D 0 if F ."/ .s/ is a proper distribution function, i.e., D 1 while ."/ > 0 if F ."/ .s/ is an improper distribution function, i.e., ."/ F .1/ < 1. Asymptotic expansions for this root play the key role in the method introduced in Chapter 2. In order to get such expansions we should impose nonlinear perturbation conditions on the distributions F ."/ .s/. These conditions mean that the power or mixed power-exponential moments of these distributions are nonlinear functions of the F ."/ .1/

10

Introduction

perturbation parameter " which can be expanded in an asymptotic power series with respect to " up to and including some order k. The method of asymptotic analysis developed in Chapter 2 is based on the combination of the generalised renewal theorem for perturbed renewal equation obtained in Chapter 1 with the asymptotic expansions for characteristic root ."/ . Using these results and some additional balancing conditions for the perturbation parameter " and time t ."/ ! 1 as " ! 0 we get the asymptotic exponential expansion of the following form: x ."/ .t ."/ / ! e r ar xQ .0/ .1/ as " ! 0: expf..0/ C a1 " C    C ar 1 "r 1 /t ."/ g

(0.12)

This asymptotic expansion is provided by the explicit effective algorithm for calculating the coefficients a1 ; : : : ; ak as functions of the coefficients in the expansions involved in the nonlinear perturbation conditions mentioned above. Despite the nonlinear and complex character of the perturbations involved we achieved our goal and have built the algorithm in an effective form of recurrence algebraic relations. We also give the asymptotical expansion for the renewal limit of nonlinearly perturbed renewal equation, R 1 s."/ ."/ e q .s/m.ds/ ."/ xQ .1/ D 0R 1 ; (0.13) ."/  s F ."/ .ds/ 0 se (here m.ds/ is the Lebesgue measure) which take the following form xQ ."/ .1/ D xQ .0/ .1/ C f1 " C    C fk "k C o."k /:

(0.14)

This expansion is also provided by the explicit recurrence algorithm for calculating the coefficients f1 ; : : : ; fk as functions of the coefficients in the expansions involved in the nonlinear perturbation conditions mentioned above. The results presented in Chapters 1 and 2, in particular asymptotics of type (0.12) and (0.14) make a basis for our studies of pseudo- and quasi-stationary phenomena for nonlinearly perturbed stochastic processes.

Chapter 3 This chapter is devoted to studies of quasi- and pseudo-stationary asymptotics for nonlinearly perturbed regenerative processes. It contains results concerning mixed ergodic and limit theorems and mixed ergodic and large deviation theorems for nonlinearly perturbed regenerative processes. These theorems are given in a form of exponential asymptotics for joint distributions of positions of regenerative processes and regenerative stopping times. These distributions satisfy some renewal equations, and the corresponding theorems are obtained by applying results of Chapters 1 and 2 to these renewal equations.

11

Introduction

."/

We consider a regenerative process ."/ .t /, t  0, with regeneration times n , n D 1; 2; : : :, and a regenerative stopping time ."/ that regenerates jointly with the process ."/ .t /, at times n."/ . Both the regenerative process ."/ .t / and the regenerative stopping time ."/ are assumed to depend on a small perturbation parameter "  0. The processes ."/ .t /, for " > 0 are considered as a perturbation of the process .0/ .t /, and therefore we assume some weak continuity conditions for certain characteristic quantities of these processes regarded as functions of " at point " D 0. As far as the regenerative stopping times are concerned, we consider two cases. The first one is where the random variables ."/ are stochastically unbounded, i.e., ."/ tend to 1 in probability as " ! 0. The second one is where the random variables ."/ are stochastically bounded as " ! 0. The object of our study is the probabilities Pf ."/ .t / 2 A; ."/ > t g and their asymptotic behaviour as t ! 1 and " ! 0. We also present results concerning conditions for convergence and finding asymptotic expansions for quasi-stationary distributions of nonlinearly perturbed regenerative processes. Two types of characteristic quantities that are related to one regeneration period play a crucial role in our analysis. These are the stopping probability f ."/ D Pf."/ < 1."/ g ."/ n ."/  ."/ /, or and the power moments of the regeneration times, m."/ n D E. 1 / . 1 ."/

."/

."/

their mixed power-exponential analogues  ."/ .; n/ D E. 1 /n e 1 .."/  1 / up to the order n D k. One of the main new elements in this study is that we consider a model with nonlinear perturbations. This means that the stopping probabilities f ."/ ."/ and the moments mn or  ."/ .; n/ are nonlinear functions of ". However, we restrict our attention to the case where the nonlinear perturbations are smooth and assume that these functions can be expanded in a power series with respect to " up to and including the order k. It turns out that the relationship between the rates with which " tends to zero and the time t tends to infinity is important for the results. The balance between the rate of perturbation and the rate of growth of time is characterised by the following condition controlled by an integer parameter r, 1  r  k and taking the form of asymptotic relation "r t ."/ ! r < 1 as " ! 0. As was mentioned above, the probabilities x ."/ .t; A/ D Pf ."/ .t / 2 A; ."/ > t g satisfy the renewal equation with the forcing function q ."/ .t; A/ D Pf ."/ .t / 2 A;

1."/ > t; ."/ > t g and the distribution function F ."/ .s/ D Pf 1."/  s; ."/  1."/ g generating the corresponding renewal equation. This let us write down the asymptotic exponential expansion for the above probabilities that replicates the asymptotic expansion in the relation (0.12) for the probabilities x ."/ .t; A/. We also write down the asymptotic expansion for the the corresponding renewal limits xQ ."/ .1; A/ and then for the quasi-stationary distributions  ."/ .A/ D xQ ."/ .1; A/=xQ ."/ .1; X/ that follows from the asymptotic expansion (0.14).

12

Introduction

Finally, we expand this asymptotics to the model of nonlinearly perturbed regenerative processes with transition period. The asymptotic results obtained in Chapters 1–3 play the key role in further studies. We apply them for getting asymptotic expansions in mixed ergodic and large deviation theorems for nonlinearly perturbed semi-Markov processes with absorption. We make a very detailed analysis of pseudo- and quasi-stationary phenomena for perturbed semi-Markov processes with a finite set of states and absorption in the next two chapters.

Chapter 4 This chapter contains results concerning mixed ergodic theorems (for semi-Markov processes) and limit/large deviation theorems (for absorption times) for nonlinearly perturbed semi-Markov processes. We consider semi-Markov processes with a finite set of states X D f0; 1; : : : ; N g and the absorption state 0. The first hitting time to this state is the absorption time for the corresponding semi-Markov process. The results mentioned above are given in the form of an asymptotics for joint distributions of positions of semi-Markov processes and absorption times. They are obtained by applying the corresponding results for regenerative processes given in Chapter 3. It can be done by using the fact that a semi-Markov process can be considered as a regenerative process with regeneration times which are subsequent return moments to any fixed state i ¤ 0. The first hitting time to the absorption state 0 is, in this case, the regenerative stopping time. We consider in details not only the generic case where the limiting semi-Markov process has one communication class of recurrent-without absorption states, but also the case, where the limiting semi-Markov process has one communication class of recurrent-without absorption states and, additionally, the class of non-recurrentwithout absorption states. The latter model covers a significant part of applications. We consider a semi-Markov process ."/ .t /, t  0, with a phase space X D ."/ f0; 1; : : : ; N g and transition probabilities Qij .u/. The semi-Markov process ."/ .t / is assumed to depend on a perturbation parameter "  0. The processes ."/ .t /, for " > 0, are considered as perturbations of the process .0/ .t /, and therefore we assume some weak continuity conditions satisfied by transition quantities, namely, by the moment functionals of transition probabilities, if regarded as functions of ", at the point " D 0. ."/ Our studies are concerned with the random functional 0 which is the first hitting time for the process ."/ .t / to hit the absorption state 0. In applications, the hitting times ."/ 0 are often interpreted as transition times for different stochastic systems described by semi-Markov processes; these are occupation times or waiting times in queueing systems, lifetimes in reliability models, extinction times in population dynamic models, etc.

13

Introduction ."/

The object of our study are probabilities Pi f."/ .t / D j; 0 > t g and their asymptotic behaviour for t ! 1 and " ! 0. We give asymptotic relations for these probabilities based on minimal conditions ."/ that involve only first moments of transition probabilities Qij .u/ as well as asymptotic relations involving conditions imposed on exponential moments of these transition probabilities. We also formulate and prove a number of auxiliary propositions dealing with convergence of hitting probabilities, distributions, expectations, and exponential moments of hitting times. Finally, we give solidarity propositions that clarify the role of initial state in the corresponding asymptotic relations as well as solidarity properties of cyclic absorption probabilities, and moment generating functions for hitting times for perturbed semi-Markov processes. They show that the corresponding conditions for convergence formulated in terms of cyclic characteristics of return times, and the form of the corresponding asymptotic relations are invariant with respect to the choice of the recurrent-without-absorption state used to construct the regeneration cycles. ."/ In this case, the the distribution function 0 Gi i .t / of the return-without-absorption time in a state i ¤ 0 generates the corresponding renewal equation and the correR1 ."/ sponding characteristic equation takes the form 0 e s 0 Gi i .ds/ D 1. In particular, we show that the characteristic root ."/ of this equation does not depend on the choice of a recurrent-without-absorption state i ¤ 0. The results obtained in Chapter 4 play a preparatory role. The mixed ergodic and large deviation theorems obtained in this chapter, together with the corresponding solidarity propositions for cyclic characteristics deduced for perturbed semi-Markov processes, create a basis for getting exponential expansions in the mixed ergodic and large deviation theorems. Such expansions are given in the next chapter.

Chapter 5 In this chapter, we continue studies began in Chapter 4. We give asymptotic results which improve the mixed ergodic theorems (for semi-Markov processes) and large deviation theorems (for absorption times) to the form of exponential expansions for joint distributions of positions of semi-Markov processes and absorption times. We also give asymptotic expansions for quasi-stationary distributions of nonlinearly perturbed semi-Markov processes. These asymptotic expansions are obtained by applying the corresponding results, given in Chapter 3, on the exponential expansions for regenerative processes, and mixed ergodic and large deviation theorems obtained in Chapter 4 for nonlinearly perturbed semi-Markov processes. We consider the same model of perturbed semi-Markov processes as in Chapter 4. However, in Chapter 5, we impose stronger perturbation conditions on the transition ."/ probabilities of the imbedded Markov chains pij and the semi-Markov processes

14

Introduction

."/

."/

Qij .u/. We assume that these transition probabilities pij and the power or the mixed ."/

power-exponential moments of transition probabilities Qij .u/ (which are improper distribution functions in argument u) can be expanded in a power series with respect to " up to and including the order k. As above, the relationship between the rate with which " tends to zero and the time t tends to infinity has a delicate influence upon the results. The balance between the rate of perturbation and the rate of growth of time is characterised by the following asymptotic relation "r t ."/ ! r < 1 as " ! 0 that is assumed to hold for some 1  r  k. With the above assumptions and under some natural additional non-periodicity and ."/ Cram´er type conditions for the transition probabilities Qij .u/, we obtain the asymptotic relation ."/

Pi f."/ .t ."/ / D j; 0 > t ."/ g e ..0/ Ca1 "CCar1 "r1 /t ."/

.0/ ! QQ ij ..0/ /e r ar as " ! 0; i; j ¤ 0:

(0.15)

There are to alternatives in this relation, where the characteristic root (1) .0/ D 0 and (2) .0/ > 0. The first alternative (1) holds if an absorption in 0 is impossible for the limiting process .0/ .t /. In such a case, the absorption times ."/ 0 tend in probability to 1 as " ! 0. The asymptotic relation (0.15) describes pseudo-stationary phenomena for perturbed semi-Markov processes. The second alternative (2) holds if an absorption in 0 is possible for the limiting ."/ process .0/ .t /. In this case, the absorption times 0 are stochastically bounded as " ! 0. The asymptotic relation (0.15) describes, in this case, quasi-stationary phenomena for perturbed semi-Markov processes. .0/ In the pseudo-stationary case (1), the limiting coefficients QQ ij .0/ in the asymptotic relation (0.15), as expected, take the form fi.0/ Qj.0/ , where Qj.0/ are the stationary

probabilities of the limiting process .0/ .t / and fi.0/ is the probability of hitting the process .0/ .t / to the set of recurrent-without-absorption states before absorption, under condition that .0/ .0/ D i ¤ 0. The situation is not so simple in the quasi-stationary case (2). In this case, under some natural conditions, there exists a so-called quasi-stationary distribution for the semi-Markov process ."/ .t / given by the formulas j."/ .."/ / D P

."/ QQ kj .."/ /

QQ ."/ .."/ / r ¤0  kr

;

j ¤ 0:

."/ In this case, the coefficients QQ ij .."/ /, which are defined as in (0.15) but for a fixed "  0, may depend on the initial state i ¤ 0. But, the quasi-stationary probabilities

Introduction

15

."/

j .."/ /; j ¤ 0, do not depend on the choice of the state k ¤ 0 in the formulas above. We also give the following asymptotic expansions for the quasi-stationary distributions for nonlinearly perturbed semi-Markov processes with absorption, j."/ .."/ / D j.0/ ..0/ / C gj Œ1" C    C gj Œk"k C o."k /; j ¤ 0:

(0.16)

Both asymptotic expansions (0.15) and (0.16) are provided by the explicit algorithms for calculating the coefficients in these expansions. We obtain mixed large deviation and ergodic theorems for nonlinearly perturbed semi-Markov processes using exponential expansions in mixed ergodic and large deviation theorems developed in Chapter 3 for nonlinearly perturbed regenerative processes. We use the fact that any semi-Markov processes is a regenerative process with the return times to some fixed state being regeneration times. The time of absorption (the first time of hitting the absorption state 0) is a regenerative stopping time. A semi-Markov process is characterized by prescribing its transition probabilities. It is natural to formulate the perturbation conditions in terms of these transition probabilities. However, the conditions and expansions formulated for regenerative processes are specified in terms of expansions for the moments of regeneration times. Therefore, the corresponding asymptotic expansions for absorption probabilities and the moments of return and hitting times for perturbed semi-Markov processes must be developed. Thus, as the first step, we construct asymptotic expansions for hitting probabilities, power and mixed power-exponential moments of hitting times using a procedure that is based on recursive systems of linear equations for hitting probabilities and moments of hitting times. These moments satisfy recurrence systems of linear equations with the same perturbed coefficient matrix and the free terms connected by special recurrence systems of relations. In these relations, the free terms for the moments of a given order are given as polynomial functions of moments of lower orders. This permits to build an effective recurrence algorithm for constructing the corresponding asymptotic expansions. Each sub-step in this recurrence algorithm is of a matrix but linear type, where the solution of the system of linear equations with nonlinearly perturbed coefficients and free terms should be expanded in asymptotic series. We also provide these expansions with a detailed analysis of their pivotal properties. These results have, as we think, their own values and possible applications beyond the problems studied in the book. As soon as asymptotic expansions for moments of return-without-absorption times are constructed, the second nonlinear but scalar step of construction asymptotic expansions expansions for the characteristic root ."/ can be realised. The separation of two steps described above, the first one matrix and recurrence but linear and the second one nonlinear but scalar, significantly simplify the whole algorithm.

16

Introduction

Asymptotic expansion for the characteristic root ."/ let us combine it with asymp."/ g given in Chapter 4 and totic relations for probabilities Pi f."/ .t ."/ / D j; ."/ 0 > t obtain the exponential expansion (0.15). ."/ The asymptotic expansion (0.16) for quasi-stationary distributions j .."/ / requires two more steps, which are needed for constructing asymptotic expansions at the point ."/ for the corresponding moment generation functions, giving expressions for quasi-stationary probabilities in the quotient form, and for transforming the corresponding asymptotic quotient expressions to the form of power asymptotic expansions. As a result, we get an effective algorithm for a construction of the asymptotic expansions given in relations (0.15) and (0.16). We think that the method used for obtaining the expansions mentioned above has its own value and great potential for future studies. As a particular but important example, we also consider the model of nonlinearly perturbed continuous time Markov chains with absorption. In this case, it is more natural to formulate the perturbation conditions in terms of generators of the perturbed Markov chains. Here, an additional step in the algorithms is needed, since the initial perturbation conditions for generators must be expressed in terms of the moments for the corresponding semi-Markov transition probabilities. Then, the basic algorithms obtained for nonlinearly perturbed semi-Markov processes can be applied. Chapters 1–5 present a theory that can be applied in studies of pseudo- and quasistationary phenomena in nonlinearly perturbed stochastic systems.

Chapter 6 This chapter deals with applications of the results obtained in Chapters 1–5 to an analysis of pseudo- and quasi-stationary phenomena in nonlinearly perturbed stochastic systems. Examples of stochastic systems under consideration are queueing systems, epidemic, and population dynamics models with finite lifetimes. In queueing systems, the lifetime is usually the time at which some kind of a fatal failure occurs in the system. In epidemic models, the time of extinction of the epidemic in the population plays the role of the lifetime, while in population dynamics models, the lifetime is usually the extinction time for the corresponding population. We selected several classical models being the subject of long term research studies. These models serve nowadays mainly as platforms for demonstration of new methods and innovation results. Our goal also is to show what kind of new types results related to quasi-stationary asymptotics can be obtained for such models with nonlinearly perturbed parameters. We give all types of asymptotic results studied in Chapters 1–5. They include the following: (i) mixed ergodic theorems (for the state of the system) and limit theorems (for the lifetimes) that describe transition phenomena; (ii) mixed ergodic and large deviation theorems that describe pseudo- and quasi-stationary phenomena; (iii)

Introduction

17

exponential expansions in mixed ergodic and large deviation theorems; (iv) theorems on convergence of quasi-stationary distributions; and (v) asymptotic expansions for quasi-stationary distributions. In all the examples, we try to specify and to describe, in a more explicit form, conditions and algorithms for calculating the limit expressions, the characteristic roots, and the coefficients in the corresponding asymptotic expansions. As the first example, we consider a M/M queueing system with highly reliable main servers. The simplest variant of such system with nonlinearly perturbed parameters was the subject of discussion in the first three subsections above. This queueing system is our first choice because of its function can be described by some nonlinearly perturbed continuous time Markov chains with absorption. Here, all conditions take a very explicit and clear form. We also consider M/G queueing system with quick service and a bounded queue buffer. In this case, the perturbed stochastic processes, which describe the dynamics of the queue in the system, belong to the class of so-called stochastic processes with semi-Markov modulation. These processes admit a construction of imbedded semiMarkov processes and are more general than semi-Markov processes. This example was chosen because it shows in which way the main results obtained in the book can be applied to stochastic processes more general than semi-Markov processes, in particular, to stochastic processes with semi-Markov modulation. Our next example is based on classical semi-Markov and Markov birth-and-death type processes. Some classical models of queueing systems, epidemic or population dynamic models can be described with the use of such processes. We show in which way nonlinear perturbation conditions should be used and what form will take advanced quasi- and pseudo-stationary asymptotics developed in Chapters 1–5. Finally, we consider an example of nonlinearly perturbed metapopulation model. We think that this example is interesting since it brings, for the first time, the discussion on advanced quasi- and pseudo-stationary asymptotics in this actual area of research in mathematical biology.

Chapter 7 The classical risk processes are still the object of intensive research studies as show, for example, references given in the bibliography. Of course, the purpose of these studies is not any more to derive formulas relevant for field applications. These studies intend to illustrate new methods and types of results that can later be expanded to more complex models. The same approach was used by us when choosing this model. The aim was to show that the innovative methods of analysis for nonlinearly perturbed processes developed in the book can yield new results for this classical models. This chapter contains results that extend the classical Cram´er–Lundberg and diffusion approximations for the ruin probabilities to a model of nonlinearly perturbed risk

18

Introduction

processes. Both approximations are presented in a unified way using the techniques of perturbed renewal equations developed in Chapters 1 and 2. The main new element in our results is a high order exponential asymptotic expansion in these approximations for nonlinearly perturbed risk processes. Correction terms are obtained for the Cram´er–Lundberg and diffusion type approximations, which provide the right asymptotic behaviour of relative errors in the perturbed model. We study the dependence of these correction terms on the relations between the rate of perturbation and the rate of growth of the initial capital. We also give various variants of the diffusion type approximation, including the asymptotics for increments and derivatives of the ruin probabilities. Finally, we give asymptotic expansions in the Cram´er–Lundberg and diffusion type approximations for distribution of the capital surplus prior and at ruin for nonlinearly perturbed risk processes. It seems to us that results presented in Chapters 6 and 7 illustrate well a potential of asymptotic methods developed in the book.

Chapter 8 The last chapter contains three supplements. The first one gives some basic arithmetic operation formulas for scalar and matrix asymptotic expansions. In the second supplement, we discuss some new prospective directions for future research studies in the area and make comments on some already available results. Let us point out some of these directions. In the book, we study quasi-stationary phenomena for nonlinearly perturbed stochastic processes and systems with continuous time. However, it is clear that similar results can also be obtained for discrete time models which are important by themselves and in applications. Another direction for future studies is the exponential asymptotic expansions, which are based on non-polynomial systems of infinitesimals, in mixed ergodic and largedeviation theorems for nonlinearly perturbed regenerative, Markov, and semi-Markov processes. In the book, we treat models of nonlinearly perturbed Markov chains and semiMarkov processes with a finite phase space. It would be important to realise an analogous program of studies for Markov chains and semi-Markov type processes with countable and general phase spaces. We have no doubts that the theory developed in the book can be also be successfully applied to more general stochastic processes with semi-Markov modulation (switchings). Another direction for advancing the studies carried out in the book is connected with models of nonlinearly perturbed Markov chains and semi-Markov processes with asymptotically uncoupled phase spaces.

Introduction

19

Explicit estimates for the remainder terms in the exponential asymptotic expansions for nonlinearly perturbed stochastic systems also make a natural object for future studies. Asymptotic analysis of pseudo- and quasi-stationary phenomena in nonlinearly perturbed queueing systems and networks, which can be described with the use of Markov chains, semi-Markov processes, and stochastic processes with semi-Markov modulation, is an unlimited area for applications of the theory developed in the book. Examples given in the book make just the first step in exploring this unlimited area. Nonlinearly perturbed epidemic and population dynamics models with different types, e.g., sex, age, health status, etc., of individuals in the population, forms of interactions between individuals, environmental modulation, complex spatial structure, etc., also make interesting objects for asymptotic analysis of pseudo- and quasi-stationary phenomena. In the book the methods of asymptotic analysis of nonlinearly perturbed renewal equations are also applied to classical risk processes. There are no doubts that these methods can be also applied to more general risk type processes, for example, risk processes with investment components described by Brownian motions, more general L´evy type risk processes, risk processes with modulated claim flows, etc. Finally, we would like also to mention a possible prospective area of financial applications of asymptotic methods developed in the book that is Markov and semi-Markov processes with absorption describing credit rating dynamics. We hope that the remarks and the analysis of new directions of research presented in Chapter 8 will be especially useful for young researchers and stimulate their interest to research studies in these areas. The book contains also extended and carefully gathered bibliography that has more than 1000 references to works in the areas related to the subject of the book. The last supplement in Chapter 8 contains the corresponding brief bibliographical remarks.

Conclusion Quasi-stationary phenomena and related problems are a subject of intensive studies during several decades. However, the development of theory of quasi-stationary phenomena is still far from its completion. The part of the theory related to conditions of existence of quasi-stationary distributions is comparatively well developed while computational aspects of the theory are underdeveloped. The content of the book is concentrated in this area. The book presents new effective methods for asymptotic analysis of pseudo- and quasi-stationary phenomena for nonlinearly perturbed stochastic processes and systems. Moreover, the results presented in the book unite, for the first time, research studies of pseudo- and quasi-stationary phenomena in the frame of one theory. We also think that methods of asymptotic analysis for nonlinearly perturbed stochastic processes and systems developed in the book have their own values and possible applications beyond the problems studied in the book.

20

Introduction

The results presented in the book will be interesting to specialists, who work in such areas of the theory of stochastic processes as ergodic, limit, and large deviation theorems, analytical and computational methods for Markov chains, regenerative, Markov, semi-Markov, risk and other classes of stochastic processes, renewal theory, and their queueing, reliability, population dynamics, and other applications. We hope that the book will also attract attention of those researchers, who are interested in new analytical methods of analysis for nonlinearly perturbed stochastic processes and systems, especially those who like serious analytical work. This gives us hope that the book will find sufficient number of readers interested in stochastic processes and their applications.

Chapter 1

Perturbed renewal equation

Chapter 1 contains results that generalise the classical renewal theorem to the model of the perturbed renewal equation. The subject of our studies is the following classical renewal equation on the halfline: Z t x.t / D q.t / C x.t  s/F .ds/; t  0: (1.0.1) 0

The so-called renewal theorem given in its final form by Feller (1966) is one of the main results in the renewal theory. This theorem gives conditions for existence of a limit of x.t / as t ! 1. It states that if (a) the distribution F .s/ in non-arithmetic and has a finite mean, and (b) the function q.t / is a directly Riemann integrable on Œ0; 1/, then x.t / ! x.1/ as t ! 1; (1.0.2) and it also gives an explicit expression for the limit x.1/ in terms of q.s/ and F .s/. The renewal theorem is a very powerful tool for proving ergodic theorems for regenerative stochastic processes. This class of processes is very broad. It includes Markov processes with discrete phase space. Moreover, any Markov process with a general phase space can be included in a model of regenerative processes with the help of the procedure of artificial regeneration. Applying the renewal theorem to ergodic theorems for regenerative type processes is based on the well-known fact that the distribution of a regenerative process at a moment t satisfies a renewal equation. This makes it possible to apply the renewal theorem and to describe the asymptotic behaviour of the distribution of the regenerative process as t ! 1. Some applications, for example when one studies the so-called transition phenomena for stochastic processes, lead to a study of more general ergodic theorems in the so-called triangular array mode. In such models, the characteristics of the stochastic process depend on some perturbation parameter, and one studies the asymptotic behaviour of the distributions of the process as t ! 1 and the perturbation parameter tends to some limiting value, simultaneously. This problem leads, in a natural way, to generalisations of the renewal theorem to a model with a perturbed renewal equation, Z t x ."/ .t / D q ."/ .t / C x ."/ .t  s/F ."/ .ds/; t  0: (1.0.3) 0

22

1

Perturbed renewal equation

In this model, the forcing function q ."/ .t / and the distribution F ."/ .s/ depend on some perturbation parameter "  0 and converge, in some natural sense, to q .0/ .t / and F .0/ .s/ as " ! 0, correspondingly. These continuity conditions allow to consider equation (1.0.3) for " > 0 as perturbations of equation (1.0.3) for " D 0. In the perturbed model, the solution x ."/ .t / of the renewal equation (1.0.1) also depends on the parameter ", and the problem is to describe the asymptotic behaviour of the solution x ."/ .t / in the so-called triangular array mode when t ! 1 and " ! 0, simultaneously. Without loss of generality, one can assume that the time t D t ."/ is a function of the perturbation parameter ". It is also important to generalise the renewal theorem to the model of the perturbed renewal equation in such a way so that, in the case of the non-perturbed equation, the conditions should reduce to the conditions of the classical renewal theorem mentioned above. These results are presented in Chapter 1. The most interesting asymptotics appears in the case where the distributions F ."/ .s/ may be improper, i.e., F ."/ .1/  1 but for the defect of this distribution, f ."/ D 1  F ."/ .1/, we have f ."/ ! 0 as " ! 0. In this case, the condition balancing the rate at which the time t D t ."/ approaches infinity and the convergence rate of f ."/ to zero as " ! 0 has a delicate influence upon the asymptotics of the solution x ."/ .t /. This balancing condition has the form f ."/ t ."/ !  as " ! 0:

(1.0.4)

Assuming conditions analogous to those in the renewal theorem and the balancing condition (1.0.4) to hold, we obtain an asymptotic relation that is more general than (1.0.2), x ."/ .t ."/ / ! x .0/ .1/ as " ! 0; ."/ ."/ expf t g

(1.0.5)

where " is a nonlinear functional of the distributions F ."/ .s/ such that " ! 0 as " ! 0. Moreover, it can be shown that ."/

."/  f ."/ =m1

as " ! 0;

(1.0.6)

."/ .s/. where m."/ 1 is the mean of the distribution F It should be noted that this asymptotics also covers the case of proper perturbed renewal equations when f ."/  0 for all "  0. In this case, the balancing condition (1.0.4) holds automatically with  D 0 if t ."/ ! 1 in an arbitrary way. However, in the general case, the balancing condition (1.0.4) imposes a restriction on the rate of growth of time. This restriction becomes unnecessary if an additional Cram´er type condition is imposed on the distributions F" .s/. In this case, it can be shown that the asymptotic

1.1

Renewal equation

23

relation (1.0.5) holds without any restriction on t ."/ ! 1 with the coefficient ."/ given as solution of the characteristic equation Z

1 0

."/

e s F ."/ .ds/ D 1:

(1.0.7)

Moreover, in this case the defect f ."/ can be asymptotically separated from zero, i.e., the limiting renewal equation can also be improper. The main results of the Chapter 1 are Theorems 1.2.1, 1.3.1, 1.4.1 and 1.4.2. Theorem 1.2.1 is a direct generalisation of the classical renewal theorem to the model of the perturbed renewal equation. The conditions of this theorem are reduced to the conditions of the classical renewal theorem in the case of the non-perturbed equation, so that it is a generalisation obtained without technical losses in the conditions. Theorems 1.3.1 and 1.4.1 extend the generalisations of the renewal theorem to the model of the improper perturbed renewal equation, where the distributions generating the perturbed renewal equation could have the total variation less than 1 but tending to 1. The deficiency of the distribution of the generating renewal equation causes an appearance of additional exponential normalising functions and additional multiplicative factors in the renewal limits. There is a principal difference between these two theorems. Theorem 1.3.1 uses some condition that balances the rate of growth of time with the rate of convergence of the deficiency of the distributions, which generate the perturbed renewal equation, to zero. In Theorem 1.4.1 there are no restrictions on the rate of growth of time but some Cram´er type condition is imposed on the distribution that generates the perturbed renewal equation. Finally, Theorem 1.4.2 generalises Theorem 1.4.1 to the case where the distributions generating the perturbed renewal equation can have the total variation tending to some limit which may be less, equal or greater than 1. Theorems 1.2.1, 1.3.1, 1.4.1 and 1.4.2 are basic for both developing exponential asymptotical expansions for the nonlinearly perturbed renewal equation and an analysis and studies of mixed ergodic and large deviation theorems for nonlinearly perturbed stochastic processes and quasi-stationary phenomena in stochastic systems. A further improvement would be obtaining more explicit representations for the exponential normalising functions in (1.0.5) than those given by the characteristic equation (1.0.7). This problem is discussed in Chapter 2.

1.1 Renewal equation In this section we formulate some basic facts from the renewal theory. We refer to the classical book by Feller (1966), where one can find a detailed exposition and proofs of these facts.

24

1

Perturbed renewal equation

1.1.1 Renewal equation Let F .x/ be a distribution function on Œ0; 1/ which is not concentrated at 0. Let also q.t / be a function from the class L of measurable (Borel) real-valued functions defined on Œ0; 1/ and bounded on any finite interval. The following equation, known as the renewal equation, plays a fundamental role in the renewal theory and its applications: Z t x.t / D q.t / C x.t  s/F .ds/; t  0: (1.1.1) 0

Rt Note that here and in the sequel the integration s is performed over the closed interval Œs; t  if 0  s  t < 1 and over the semi-closed interval Œs; 1/ if 0  s < t D 1. We will call F .x/ the distribution function generating the renewal equation (1.1.1), and q.t / the forcing function of this equation. A solution of the renewal equation (1.1.1) is sought for in the same class L that contains the forcing function q.s/. Let us denote by F .r / .t / the r-fold convolution of the distribution function F .x/, i.e., the distribution function defined by recurrence, ( for r D 0, Œ0;1/ .t /; t  0 .r / F .t / D (1.1.2) F ..r1// .t /  F .t /; t  0 for r  1, where the convolution F .t / D F1 .t /  F2 .t / of two proper or improper distribution functions F1 .t / and F2 .t /, concentrated on the non-negative half-line, is defined by the following formula: ( Rt 0 F1 .t  s/F2 .ds/ for t  0, F .t / D 0 for t < 0. The following function, called a renewal function, plays an important role in the theory: U.t / D

1 X

F .r / .t /;

t  0:

(1.1.3)

r D0

It can easily be shown that the series in (1.1.3) converges for all t  0 and, moreover, U.t /  K1 C K2 t for some finite constants K1 and K2 , so that it has a linear rate of growth in t . Also note that by the definition U.t / is a nondecreasing rightcontinuous function. We also consider the renewal measure U.A/ on the Borel -algebra of Œ0; 1/ generated by the renewal function in the usual way. This measure is defined on intervals by U..a; b/ D U.b/  U.a/, 0  a  b < 1.

1.1

25

Renewal equation

Note that this measure always has an atom in zero, since we have the 0-fold convolution F .0/ .t / in (1.1.3). This convolution is, by the definition, a distribution function that has a unit jump in the point zero. As is well known, a unique solution of the renewal equation in the class L is of the form Z t x.t / D q.t  s/U.ds/; t  0: (1.1.4) 0

1.1.2 Renewal theorem An important role in probability theory is played by the so-called renewal theorem that describes the behaviour of the solution of (1.1.1) for t ! 1. There is a long history of developing results related to this theorem. The final form and the proof of the renewal theorem was given by Feller (1966). We follow his presentation. To do this we need in some definitions. A distribution function F .x/ is arithmetic if there exists h > 0 such that 1 X

.F .rh/  F .rh  0// D 1:

(1.1.5)

r D1

If F .x/ is an arithmetic distribution function, then there exists the largest positive h such that (1.1.5) holds. This h is called a step of the arithmetic distribution function. According the above definition, a distribution function F .x/ is non-arithmetic if relation (1.1.5) does not hold for any positive h, that is, 1 X

.F .rh/  F .rh  0// < 1;

h > 0:

(1.1.6)

r D1

Formula (1.1.6) implies that a non-arithmetic distribution function can not be concentrated in zero. A measurable function q.t / is directly Riemann integrable on Œ0; 1/ if it is continuous almost everywhere with respect to the Lebesgue measure on Œ0; 1/, bounded on any finite interval, and there exists h > 0 such that X lim h sup jq.t /j D 0: (1.1.7) T !1

r T = h r ht.rC1/h

Let us remark that a function q.t / that is directly RiemannR integrable on Œ0; 1/ 1 is also absolutely integrable on Œ0; 1/ in the usual way, and T jq.s/j ds ! 0 as T ! 1. Let us denote the first moment of the distribution function F .x/ by Z 1 m1 D tF .dt /: 0

26

1

Perturbed renewal equation

The conditions for the renewal theorem are the following: D1 : F .x/ is a non-arithmetic distribution function; M1 : m1 < 1; and F1 : q.t / is a directly Riemann integrable function on Œ0; 1/. Now we are prepared to formulate the classical variant of the so-called renewal theorem in the form as was given by Feller (1966). Theorem 1.1.1. Let conditions D1 , M1 , and F1 hold. Then Z 1 1 q.s/ ds as t ! 1: x.t / ! m1 0

(1.1.8)

We refer to the book Feller (1966, 1971) for a proof of this theorem. Let us remark that, in the next section, we formulate and give a proof of a theorem that generalises the renewal theorem to the so-called triangular array mode.

1.1.3 Renewal process We also need to formulate some additional facts concerning the so-called renewal processes. Let k , k D 1; 2; : : :, be a sequence of nonnegative independent identically distributed (i.i.d.) random variables with a distributionP function F .x/ that does not n have zero in its support. The random variables n D kD1 k , n D 0; 1; : : :, can be interpreted as renewal times and a renewal process can be defined as .t / D max.n W n  t /, t  0. By definition, .t / is the number of renewals in the interval Œ0; t . The following well-known formula expresses the renewal functions as the expectation of a renewal process, U.t / D E.t / C 1;

t  0:

(1.1.9)

We use the following notation: .t / D

.t/C1 X

k  t;

t  0:

kD1

By definition, .t / is the time between the moment t and the first after t renewal moment. The processes .t / and .t / are connected by the following relation: Pf.t C h/  .t / D ng D ı.n; 0/Pf.t / > hg Z h Pf.h  s/ C 1 D ngPf.t / 2 dsg; C 0

n D 0; 1; : : : ;

(1.1.10)

1.1

27

Renewal equation

where ı.r; k/ is the Kronecker delta-function that is ı.r; k/ D 1 if r D k and ı.r; k/ D 0 if r ¤ k. Relation (1.1.10) implies the following formula: E..t C h/  .t //r D

Z

h 0

E..h/ C 1/r Pf.t / 2 dsg;

r D 1; 2; : : : : (1.1.11)

The following lemma gives a useful upper bound for the moments of the random variables .t /. Lemma 1.1.1. If .˛/ F .g/  G for some g > 0 and G < 1, then for every r D 1; 2; : : : and t  0, E.t /r  .t C g/r KŒr; g; G; where KŒr; g; G D g r

1 X

(1.1.12)

.n C 1/r G n < 1:

nD0

Proof. Relation (1.1.11) obviously implies the following inequalities: E..t C h/  .t //r  E..h/ C 1/r ;

h; t  0;

r D 1; 2; : : : :

(1.1.13)

Using inequalities (1.1.13) we get r

E.t / D E

 Œt=g X

r ..kg/  ..k  1/g// C ..t /  .Œt =gg//

kD1 r 1

 .Œt =g C 1/

 Œt=g X

r E..kg/  ..k  1/g/

kD1 r

C E..t /  .Œt =gg/ 



t Cg g

r

E..g/ C 1/r :

Under the condition of Lemma 1.1.1,  X n

k1  g  F .g/n  G n ; Pf.g/  ng D P

n D 0; 1; : : : :

(1.1.14)

(1.1.15)

kD1

It follows from (1.1.15) that E..g/ C 1/r  LŒr; G D

1 X

.n C 1/r G n < 1:

nD0

Now, inequality (1.1.12) follows from (1.1.14) and (1.1.16).

(1.1.16)

28

1

Perturbed renewal equation

Lemma 1.1.2. If condition .˛/ given in Lemma 1.1.1 holds, then for every r D 1; 2; : : : and h  0, sup E..t C h/  .t //r  LŒh; r; g; G;

(1.1.17)

t0

where LŒh; r; g; G D 2r 1 ..h C g/r KŒr; g; G C 1/ < 1: Proof. Using inequality (1.1.13) and Lemma 1.1.1 we get sup E..t C h/  .t //r  E..h/ C 1/r  2r1 .E.h/r C 1/ t0

 2r 1 ..h C g/r KŒr; g; G C 1/:

(1.1.18)

That proves the lemma. Let d.A/ denote the diameter of the bounded set A. Lemma 1.1.3. If condition .˛/ given in Lemma 1.1.1 holds, then for an arbitrary measurable bounded set A Œ0; 1/, sup U.t C A/  .d.A/ C 2g/LŒh; r; g; G < 1: t0

(1.1.19)

Proof. The proof follows in an obvious way from Lemma 1.1.2, the inequalities E..t C h/  .t // D U.t C h/  U.t / D U..t; t C h/;

t; h  0;

(1.1.20)

and the observation that for any bounded subset A there exists an interval of the form .t; t C h of length h D d.A/ C g such that A .t; t C h. Remark 1.1.1. It is useful to note that inequalities (1.1.19), (1.1.18), and (1.1.20) are uniform with respect to any class F of distribution functions F .x/ for which the condition F .g/  G holds with the same constants g > 0 and G < 1 for all the distribution functions F .x/ that belong to the class F.

1.2 Renewal theorem for perturbed renewal equation In this section we formulate and prove a theorem that generalises the classical renewal theorem to a model of the perturbed renewal equation.

1.2 Renewal theorem for perturbed renewal equation

29

1.2.1 Perturbed renewal equation Consider the family of renewal equations, Z t ."/ ."/ x .t / D q .t / C x ."/ .t  s/F ."/ .ds/; 0

t  0;

(1.2.1)

where, for every "  0, we have the following: (a) q ."/ .t / is a real-valued function on Œ0; 1/ that is measurable and locally bounded, i.e., bounded on every finite interval, and (b) F ."/ .s/ is a distribution function on Œ0; 1/ which is not concentrated at 0. Denote Z 1 ."/ m1 D sF ."/ .ds/: F ."/ ./

0 .0/ F ./

) as " ! 0 means weak convergence of the As usual the symbol distribution functions, that is, the pointwise convergence in each point of continuity of the limiting distribution function. We assume that the functions q ."/ .t / and the distributions F ."/ .s/ satisfy the following continuity conditions at the point " D 0, if regarded as functions of ": D2 : F ."/ ./ ) F .0/ ./ as " ! 0, where F .0/ .s/ is a non-arithmetic distribution function; ."/

.0/

M2 : m1 ! m1 < 1 as " ! 0; and F2 :

(a) limu!0 lim0"!0 supjvju jq ."/ .t C v/  q .0/ .t /j D 0 almost everywhere with respect to the Lebesgue measure on Œ0; 1/; (b) lim0"!0 sup0tT jq ."/ .t /j < 1 for every T  0; P (c) limT !1 lim0"!0 h r T = h supr ht.rC1/h jq ."/ .t /j D 0 for some h > 0.

Conditions D2 , M2 and F2 are conditions for convergence of the distribution functions F ."/ .x/ and the forcing functions q ."/ .t /, respectively, to the corresponding limiting functions F .0/ .x/ and the forcing functions q .0/ .t / as " ! 0. This allows to consider equation (1.2.1) for " > 0 as a perturbed version of equation (1.2.1) for " D 0, and to interpret the parameter " as a perturbation parameter. Let us make some useful remarks concerning conditions D2 , M2 and F2 . Remark 1.2.1. The conditions D2 , M2 and F2 reduce to the conditions of the classical renewal theorem, if F ."/ .s/  F .0/ .s/ and q ."/ .t /  q .0/ .t / do not depend on the perturbation parameter ". In particular, condition D2 reduces to condition D1 , i.e., to the assumption that F .0/ .s/ is a non-arithmetic distribution function; M2 to condition .0/ M1 , i.e., to the assumption that the expectation m1 is finite; and F2 to condition F1 , i.e., to the assumption that the function q .0/ .t / is directly Riemann integrable on Œ0; 1/.

30

1

Perturbed renewal equation

Remark 1.2.2. Under condition D2 , condition M2 is equivalent to the following condition: R1 M3 : limT !1 lim"!0 T .1  F ."/ .v//dv D 0. This proposition is a direct corollary of the following well-known formula for expectations of non-negative random variables: Z 1 ."/ .1  F ."/ .v//dv: (1.2.2) m1 D 0

Indeed, let us denote Z T Z ."/ ."/ ."/ mC .T / D .1  F .v//dv; m .T / D 0

1

T

."/

.1  F ."/ .v//dv; T > 0:

."/

It is clear that m1 D mC .T / C m."/  .T /, T > 0. Conditions D2 and M2 imply, in

."/ ."/ .0/ .0/ .0/ an obvious way, that (a) m."/  .T / D m1  mC .T / ! m1  mC .T / D m .T /

.0/ .0/ as " ! 0, T > 0. Also, (b) m.0/  .T / D m1  mC .T / ! 0 as T ! 1. Relations (a) and (b) imply condition M3 . On the other hand, condition D2 implies that (c) R 0 R T0 ."/ .v//dv D T .1  F .0/ .v//dv, for 0  T < lim"!0 m."/ C .T /  lim"!0 T .1  F T .0/ .T /  m T 0 < 1, and, therefore, (d) lim"!0 m."/   .T /, for 0  T < 1. Relation (d) and condition M3 imply that (e) m.0/ .T / ! 0 as T ! 1. Finally, relation (e)  ."/ .0/ ."/ and conditions D2 and M3 imply that (e) lim"!0 jm1  m1 j  lim"!0 .jmC .T /  .0/

.0/ ."/ .0/ mC .T /j C m."/  .T / C m .T // D lim"!0 m .T / C m .T / ! 0 as T ! 1. Thus, condition M2 holds. Note, that, according the proof given above, if condition D2 holds then condition M3 can be formulated in the equivalent form that let parameter " to tend to 0 being non-negative in the corresponding asymptotic relation: R1 M03 : limT !1 lim0"!0 T .1  F ."/ .v//dv D 0.

Remark 1.2.3. Denote by m.A/ the Lebesgue measure on the real line. Let also C0 be the set of convergence in the condition F2 . By this condition, m.C 0 / D 0. Condition F2 implies that every point t 2 C0 is a point of continuity of the limiting function q .0/ .t /. It follows from the assumption that 0  " ! 0, which admit the case where "  0. It can be shown that conditions F2 .a/; .b/ and imply that condition F2 .a/ holds for any point of continuity of the limiting function q .0/ .t /. From these remarks it follows that the limiting function is continuous almost everywhere with respect to the Lebesgue measure on R1 . Remark 1.2.4. Conditions F2 .b/; .c/ imply that the functions q ."/ .t / are Lebesgue integrable on the interval Œ0; 1/ for all " small enough. However, there is no guarantee that these functions are directly Riemann integrable on Œ0; 1/ for " > 0. As a matter

1.2 Renewal theorem for perturbed renewal equation

31

of fact, there is no guarantee that the functions are continuous almost everywhere with respect to the Lebesgue measure on Œ0; 1/. As far as the limiting function q .0/ .t / is concerned, condition F2 implies that this function is not only Lebesgue integrable but also directly Riemann integrable on Œ0; 1/. Indeed, as it is mentioned in Remark 1.2.3, this function is continuous almost everywhere with respect to the Lebesgue measure on Œ0; 1/. Also relation (1.1.7) given in the definition of a directly Riemann integrable function is satisfied for q .0/ .t /, since the assumption that 0  " ! 0 admits the case where "  0. This remark allows to replace the Lebesgue integral over Œ0; 1/ for this function with the corresponding Riemann integral in the left-hand side of the following equality: Z 1 Z 1 .0/ q .s/m.ds/ D q .0/ .s/ ds: (1.2.3) 0

0

1.2.2 Renewal theorem for perturbed renewal equation The main result of this section is the following theorem that generalises the classical renewal theorem to the model of perturbed renewal equation. The conditions of this theorem reduce to to the conditions of the classical renewal Theorem 1.1.1 in the case where of non-perturbed renewal equation. Theorem 1.2.1. Let conditions D2 , M2 , and F2 hold. Then for any 0  t ."/ ! 1 as " ! 0, R 1 .0/ q .s/ds ."/ ."/ .0/ x .t / ! x .1/ D 0 as " ! 0: (1.2.4) .0/ m1 The proof of this Theorem 1.2.1 is based on a generalisation of the so-called Blackwell theorem to the model of the perturbed renewal equation. Theorem 1.2.2. Let conditions D2 and M2 hold. Then for any 0  t ."/ ! 1 as " ! 0 and h  0, .0/

U ."/ .t ."/ C h/  U ."/ .t ."/ / ! h=m1

as " ! 0:

(1.2.5)

1.2.3 Convergence of Lebesgue integrals in the scheme of series We postpone the proof of Theorems 1.2.1 and 1.2.2 and, first, prove the following proposition that we shall constantly use in what follows. Let, for every "  0, f ."/ .s/ be real-valued bounded Borel functions defined on R1 . U

We use the symbol f ."/ .s/ ! f .0/ .s/ as " ! 0 to indicate that functions f ."/ ./ converge to a function f .0/ ./ locally uniformly at point s as " ! 0. This means that lim

lim sup jf ."/ .s C v/  f .0/ .s/j D 0:

0 0 such that (f) js" sj  u. / for "  ".ı/. Relations (e) and (f) imply that (g) jf ."/ .s" /f .0/ .s/j  supjvju. / jf ."/ .s C v/  f .0/ .s/j  for "  ".ı/. Since an arbitrary choice of > 0, relations (g) imply that condition .˛/ hold. Let B.1/ denote the Borel -algebra on R1 and let, for every "  0, " .A/ be a measure on B.1/ taking finite values on bounded Borel sets. We use the symbol " .A/ ) 0 .A/ as " ! 0 to indicate that the measures " .A/ weakly converge to a measure 0 .A/ as " ! 0. This means that, for all 1 < u  v < 1 such that the limiting measure has not atoms in the points u and v, ."/ ..u; v/ ! .0/ ..u; v/ as " ! 0:

(1.2.7)

Let now f ."/ .s/ be real-valued bounded Borel functions defined on R1 and ."/ .A/ be finite measures on B.1/ for every "  0. Lemma 1.2.2. Let the following conditions hold: .˛/ ."/ .A/ ) .0/ .A/ as " ! 0; .ˇ/ ."/ .R1 / ! .0/ .R1 / < 1 as " ! 0; ./ lim"!0 sups2R1 jf ."/ .s/j D f < 1; .ı/ functions f ."/ ./ converge to f .0/ ./ as " ! 0 locally uniformly at every point s 2 S , where S is some subset of B.1/ ; ./ .0/ .S / D 0. Then Z Z ."/ ."/ f .s/ .ds/ ! f .0/ .s/.0/ .ds/ as " ! 0: (1.2.8) R1

R1

Proof. If .0/ .R1 / D 0 then the statement of the lemma follows in an obvious way from conditions .ˇ/ and ./, and the corresponding limit takes the value zero. If .0/ .R1 / > 0 then, condition .ˇ/ implies that ."/ .R1 / > 0 for " small enough,

1.2 Renewal theorem for perturbed renewal equation

33

say "  "0 . Condition .ˇ/ permits to reduce the proof to the case where ."/ ./ are probability measures. This follows from the identity Z Z f ."/ .s/."/ .ds/ D ."/ .R1 / f ."/ .s/Q ."/ .ds/; R1

R1

where Q ."/ .A/ D ."/ .A/=."/ .R1 / is a probability measure and "  "0 . Also, taking the definition of a limit in terms of convergent subsequences, one can always reduce the consideration to the case where the parameter " ! "0 D 0 runs only over some subsequence of positive values, "n ; n D 1; 2; : : : such that "n ! 0 as n ! 1. The Skorokhod (1956) representation theorem (see, for example, Silvestrov (2004a), page 20) implies that, under condition .˛/, it is possible to construct a sequence of random variables ."n / defined on the same probability space .; F ; P / such that (h) the random variable ."n / has the distribution ."n / ./ for every n D a:s: 0; 1; : : :; (i) ."n / ! .0/ (almost sure) as n ! 1. Let A be the set of elementary events ! 2  such that (j) ."n / .!/ ! .0/ .!/ as n ! 1. Let also B be the set of elementary events ! for which .0/ .!/ 2 S . By (i) and condition ./ that (k) P.A \ B/ D 1. By Lemma 1.2.1, it follows from (j) and condition .ı/ that (l) f ."n / .."n / .!// ! f .0/ ..0/ .!// as n ! 1 for all a:s: ! 2 A \ B. Relations (k) and (l) mean that (m) f ."n / .."n / / ! f .0/ ..0/ / as n ! 1. Note that condition ./ implies that (n) there exists N < 1 such that jf ."n / .."n / /j  2f < 1 for n  N . Also conditions ./ and .ı/ imply that (o) sups2S jf .0/ .s/j  2f < 1, and, therefore, by condition ./, (p) jf .0/ ..0/ /j  2f < 1 with probability 1. Hence, by the Lebesgue theorem, it follows from (m), (n), and (p) that Ef ."n / .."n / / ! Ef .0/ ..0/ / as n ! 1. By (h), this is equivalent to the assertion of the lemma.

1.2.4 Proof of Theorem 1.2.2 Denote by U ."/ .t / the renewal function generated by the distribution function F ."/ .t / and defined by formula (1.1.3). It is also convenient to extend the renewal function to the negative half-line by U ."/ .t / D 0 for t < 0 and consistently to consider the corresponding renewal measure U ."/ .A/ on the Borel -algebra on R1 defined by the formula U ."/ ..u; v/ D U ."/ .v/  U ."/ .u/ for all 1 < u  v < 1. By definition, U ."/ ..1; 0// D 0. The method of the proof follows that in Feller (1966). However, triangular array aspects involved in the model cause many technical complications in the proof. Let us consider the shifted renewal measures U ."/ .t C A/, t  0 (here t C A is the set of the points x C t where x 2 A). By definition, the measure U ."/ .t C A/ has its support in the interval Œt; 1/. Since the distribution function F .0/ .t / is non-arithmetic, (q) there exists g > 0 which is a continuity point for this distribution function such that F .0/ .g/ < 1. Condition D2 implies that (r) F ."/ .g/ ! F .0/ .g/ as " ! 0. Relations (q) and (r) imply

34

1

Perturbed renewal equation

that (s) there exist "0 > 0 such that F ."/ .g/  G < 1 for 0  "  "0 . Due to Lemma 1.1.3 and Remark 1.1.1, relation (s) implies that for any measurable bounded set A R1 , lim sup U ."/ .t C A/  K1 d.A/ C K2 < 1;

(1.2.9)

0"!0 t0

where K1 and K2 are finite constants. The statement of the theorem is equivalent to the relation U ."/ .t ."/ C A/ )

1 m.0/ 1

m.A/ as " ! 0;

(1.2.10)

where m.A/ is the Lebesgue measure on a Borel -algebra B.1/ . By a suitable form of the selection theorem (see, for example, Feller (1971), page 270) it follows from (1.2.9) that, given any sequence 0 < " D "n ! 0 as n ! 1, we can choose from this sequence a subsequence "0 D "nk ! 0 as k ! 1 such that 0

0

U ." / .t ." / C A/ ) V .A/ as "0 ! 0;

(1.2.11)

where V .A/ is a measure defined on the Borel -algebra B.1/ of subsets of R1 and taking finite values on bounded Borel sets. Relation (1.2.10) will be proved if we show that the measure V .A/ D 1.0/ m.A/ m1

is independent of the choice of a sequence 0 < " D "n ! 0 as n ! 1 and the corresponding subsequence "0 D enk ! 0 as k ! 1 satisfying (1.2.11). We observe that by (1.2.9) the measure V .A/ satisfies V .A/  K1 d.A/ C K2

(1.2.12)

for any bounded set A 2 B.1/ . The first stage of the proof is to show that the measure V .A/ is proportional to the Lebesgue measure. That is, V .A/ D ˛m.A/, where ˛ is a constant and m.A/ is the Lebesgue measure on B.1/ . It is not hard to see that to achieve this it would suffice to show that for any continuous function q.t / on R1 vanishing outside the interval Œ0; a the function Z s q.s  v/V .dv/; s 2 R1 ; (1.2.13) h.s/ D sa

is a constant. Obviously, any function h.s/ defined by (1.2.13) is uniformly continuous in s 2 R1 owing to the uniform continuity of q.s/ in s. Due to (1.2.12), the function h.s/ is also bounded, namely, sup jh.s/j  Q  .K1 a C K2 /;

s2R1

(1.2.14)

1.2 Renewal theorem for perturbed renewal equation

35

where Q D sup jq.t /j < 1: t2R1

We now consider the convolution equation Z 1 h.s/ D h.s  v/F .0/ .dv/;

s 2 R1 :

0

(1.2.15)

By the Choquet–Deny (1960) lemma (see, for example, Feller (1966), page 364), if F .0/ ./ is a non-arithmetic distribution on Œ0; 1/, with zero not being in the support, equation (1.2.15) has a unique bounded uniformly continuous solution, which is a constant. Hence, owing to the above remarks, to prove that the measure V .A/ is proportional to the Lebesgue measure, it suffices to show that any function given by (1.2.13) satisfies the convolution equation (1.2.15). So, let q.s/ be a continuous function on R1 vanishing outside the interval Œ0; a. Let 0

x ." / .t / D

Z

t

0

q.t  s/U ." / .ds/;

0

t  0;

be a solution of the renewal equation x

."0 /

Z .t / D q.t / C

t 0

0

0

x ." / .t  s/F ." / .ds/;

t  0:

(1.2.16)

0

We consider below this solution in points t ." / C s for s 2 R1 . It is convenient to 0 define x ."/ .s/ D 0 for s < 0. In any case, t ." / C s ! 1 as "0 ! 0 and therefore, for 0 all "0 small enough, t ." / C s is positive. The function q.s  v/ vanishes in 0 and outside the interval Œs  a; s. So, x

."0 /

.t

."0 /

Z C s/ D Z D Z D

0

t ." / Cs 0 1 1 s sa

0

0

q.t ." / C s  v/U ." / .dv/ 0

0

0

0

q.s  v/U ." / .t ." / C dv/ q.s  v/U ." / .t ." / C dv/:

(1.2.17)

We would like to apply Lemma 1.2.2 to the integral in the right-hand side of (1.2.17). The problem is that the limiting measure V .A/ in (1.2.11) can have atoms in the endpoints of the interval Œsa; s. We, however, can always extend this interval to a larger interval Œsah1 ; sCh2  without changing the value of the integral in the righthand side of (1.2.17). It is always possible to choose h1 ; h2 > 0 in such a way that the measure V .A/ has no atoms in the endpoints of the new interval Œs  a  h1 ; s C h2 .

36

1

Perturbed renewal equation

Then we can replace the function q.sv/ with the function f .v/ D q.sv/Œsa;s .v/ 0 0 0 and to define the measures ." / .A/ D U ." / .t ." / CA\Œsah1; sCh2 /. Obviously, Z s Z 1 0 0 0 q.s  v/U ." / .t ." / C dv/ D f .v/." / .dv/: (1.2.18) 1

sa

By (1.2.11) and remarks made above, conditions .˛/ and .ˇ/ of Lemma 1.2.2 are 0 satisfied for the measures ." / .A/. It is also obvious that conditions ./–./ of this lemma are satisfied for the functions f .v/. By applying Lemma 1.2.2 to the integrals in the right-hand side of (1.2.18) and taking into consideration that the actual integration is performed over the interval Œs  a; s, we get for all s 2 R1 the following: Z t ."0 / Cs 0 0 ."0 / ."0 / C s/ D q.t ." / C s  v/U ." / .dv/ x .t Z D

0

1 1

0

f .v/." / .dv/ Z

! h.s/ D

s sa

q.s  v/V .dv/ as "0 ! 0: 0

(1.2.19)

0

Now we show that the convergence of x ." / .t ." / C s/ to h.s/ is uniform at every point s  0. The function q.s/ is uniformly continuous on R1 , since it is continuous and vanishes outside a finite interval. For c > 0, denote q .c/ D sup sup jq.s C h/  q.s/j s2R1 jhjc

Using (1.2.19) and (1.2.9) we find that 0

0

0

0

lim sup jx ." / .t ." / C s C h/  h.s/j  lim jx ." / .t ." / C s/  h.s/j 0

"0 !0 jhjc

" !0

Z C lim sup 0

" !0 jhjc

U  q .c/  lim 0

C1

1 ."0 /

" !0

0

0

jg.s C h  v/  g.s  v/jU ." / .t ." / C dv/ 0

.t ." / C Œs  a; s C c/

 q .c/  ŒK1 .a C c/ C K2  ! 0 as c ! 0:

(1.2.20)

0

We observe that the functions x.t ." / C s/ are bounded uniformly in s 2 R1 and asymptotically uniformly in "0 ! 0, more precisely, 0

0

lim sup jx ." / .t ." / C s/j 0

" !0 s2R1

0

0

0

0

sup Q  ŒU ." / .t ." / C s/  U ." / .t ." / C s  a/  lim 0 " !0 s2R1

 Q  .K1 a C K2 /:

(1.2.21)

37

1.2 Renewal theorem for perturbed renewal equation 0

0

Since x ." / .t ." / C s/ is a solution of the renewal equation, we find that

x

."0 /

.t

."0 /

C s/ D q.t

."0 /

D q.t

."0 /

Z C s/ C Z C s/ C

0

t ." / Cs 0 1 0

0

0

0

x ." / .t ." / C s  v/F ." / .dv/

0

0

0

x ." / .t ." / C s  v/F ." / .dv/:

(1.2.22)

0

It is obvious that q.t ." / C s/ ! 0 as "0 ! 0, since the function q.s/ is positive only in the interval Œ0; a. Due to (1.2.20), (1.2.21) and condition D2 , Lemma 1.2.2 can be applied to the second term in the right-hand side of (1.2.22). The function 0 0 0 x ." / .t ." / C s  v/ plays the role of the function f ." / .v/ for "0 > 0 and the function 0 h.s  v/ is the corresponding limiting function. The measure F ." / ./ plays the role of ."0 / 0 the measure  ./ for all "  0. We find, by applying Lemma 1.2.2, that x

."0 /

.t

."0 /

Z C s/ !

1 0

h.s  v/F .0/ .dv/ as "0 ! 0; s 2 R1 :

(1.2.23)

It follows from (1.2.19) and (1.2.23) that h.s/ satisfies equation (1.2.15). This proves that V .A/ D ˛m.A/, where ˛ is a constant and m.A/ is the Lebesgue measure on B.1/ . Hence, we can rewrite (1.2.11) in the following form: 0

0

U ." / .t ." / C A/ ) ˛m.A/ as "0 ! 0:

(1.2.24)

.0/

To prove the theorem, it remains to show that ˛ D 1=m1 . 0 0 If we choose q ." / .t / D 1  F ." / .t / in the renewal equation (1.2.16), then its solution is identically equal to 1, that is, Z 1D

t

0

0

.1  F ." / .t  s//U ." / .ds/:

0

(1.2.25)

0

Hence, for t ." / > T , Z 1 D

0/

0

Z D

t ."

0

T

0

0

0

.1  F ." / .t ." /  s//U ." / .ds/ 0

0

0

.1  F ." / .v//U ." / .t ." /  dv/ Z C

t ." T

0/

0

0

0

.1  F ." / .v//U ." / .t ." /  dv/:

(1.2.26)

38

1

Perturbed renewal equation

Taking into consideration (1.2.9) we find that for any T; g > 0, Z

t ." T

0/

0

0

0

.1  F ." / .v//U ." / .t ." /  dv/  lim 0

" !0

1 X

0

0

0

0

0

.1  F ." / .rg//.U ." / .t ." / C .r C 1/g/  U ." / .t ." / C rg//

r DŒT =g

 .Kg C K2 /  lim 0

" !0

Z  .K1 g C K2 / 

1 X

0

.1  F ." / .rg//

r DŒT =g

1

0

T 2g

.1  F ." / .s// ds:

(1.2.27)

Since conditions D2 and M2 imply condition M3 , by using (1.2.27) we get Z t ."0 / 0 0 0 .1  F ." / .v//U ." / .t ." /  dv/ D 0: (1.2.28) lim lim 0 T !1 " !0 T

Now we shall verify that the conditions of Lemma 1.2.2 are satisfied for the func0 0 0 0 0 tions f ." / .v/ D 1  F ." / .v/ and the measures ." / .A/ D U ." / .t ." /  A \ Œ0; T / (here A is the set of point y, where y 2 A). Indeed, (1.2.24) implies conditions .˛/ and .ˇ/ of this lemma. The functions 0 1  F ." / .v/ are bounded by 1 uniformly in "0 and v. Condition D2 implies that the 0 functions 1  F ." / .v/ converge to 1  F .0/ .v/, as "0 ! 0, at any continuity point v 0 of the function 1  F .0/ .v/. Since the functions 1  F ." / .v/ are monotone, this convergence is easily seen to be locally uniform at every continuity point of 1  F .0/ .v/. 0 Also, functions 1  F ." / .v/ are bounded. Since the set D0 of discontinuity points of the function 1  F .0/ .v/ is at most countable, ˛m.R0 / D 0. So, conditions ./–./ of Lemma 1.2.2 are also satisfied. 0 0 Applying Lemma 1.2.2 to the functions 1  F ." / .v/ and to the measures ." / .A/, we obtain the relation Z T 0 0 0 .1  F ." / .v// U ." / .t ." /  dv/ 0

Z !

T

0

.1  F .0/ .v//˛m.dv/ as "0 ! 0;

(1.2.29)

which holds for any T > 0. From (1.2.26), (1.2.28) and (1.2.29) it obviously follows that Z T 0 0 0 1 D lim lim .1  F ." / .v//U ." / .t ." /  dv/ 0 T !1 " !0 0 Z T

D lim

T !1 0

.0/

.0/

.1  F .0/ .v//˛m.dv/ D ˛m1 :

Hence, ˛ D 1=m1 and the proof of the theorem is completed.

(1.2.30)

39

1.2 Renewal theorem for perturbed renewal equation

1.2.5 Proof of Theorem 1.2.1 0

According to formula (1.1.3), for t ." / > T we have

x

."0 /

.t

."0 /

Z /D Z D

t ."

0/

0

q.t  s/U ." / .ds/

0 T 0

0

0

0

q ." / .v/U ." / .t ." /  dv/ Z C

t ."

0/

0

0

0

q ." / .v/U ." / .t ." /  dv/:

T

(1.2.31)

By Lemma 1.1.3, for any interval I D Œa; b of length h D b  a, 0

lim sup U ." / .t C I /  K1 C K2 h < 1; 0

" !0 t0

where K1 ; K2 < 1. Therefore, using of the condition F2 , we find for h > 0 in this condition that Z

t ."

0/

lim

"0 !0 T

 lim 0

" !0

0

0

0

jq ." / .v/jU ." / .t ." /  dv/ X

0

jq ." / .t /j

sup

r ŒT = h r ht.rC1/h 0

0

0

0

 .U ." / .t ." /  rh/  U ." / .t ." /  .r C 1/h// X K1 C K2 h 0  lim h sup jq ." / .t /j ! 0 as T ! 0:  0 " !0 h T h r ht.rC1/h r

(1.2.32)

h

Theorem 1.2.2 and condition F2 allow to apply Lemma 1.2.2 to the functions 0 0 0 0 q ." / .v/ and the measures ." / .A/ D U ." / .t ." /  A \ Œ0; T /, which yields the following relation: Z 0

T

q

."0 /

.t

."0 /

 s/U

."0 /

Z .ds/ D !

T 0

1 .0/

m1

0

0

0

q ." / .v/U ." / .t ." /  dv/ Z 0

T

q .0/ .v/m.dv/ as "0 ! 0:

(1.2.33)

40

1

Perturbed renewal equation

Relations (1.2.31), (1.2.32) and (1.2.33) imply in an obvious way that lim x

."0 /

"0 !0

.t

."0 /

Z / D lim 0

t ."

0/

0

Z

D lim lim 0

T

T !1 " !0 0

D lim

Z

1 .0/ m1 Z 1

T !1

D

1 m.0/ 1

0

0

q ." / .t ." /  s/U ." / .ds/

" !0 0

T 0

0

0

0

q ." / .t ." /  s/U ." / .ds/ q .0/ .v/m.dv/

q .0/ .v/m.dv/:

0

(1.2.34)

The limit in (1.2.34) does not depend on the choice of the sequence 0 < " D "n ! 0 as n ! 1 and the corresponding subsequence "0 D "nk ! 0 as k ! 1 for which (1.2.11) is satisfied. Thus relation (1.2.4) in Theorem 1.2.1 holds for any " ! 0, and the proof is completed. As it follows from the proof, Theorem 1.2.1 is in some sense a corollary of Theorem 1.2.2. At the same time, it is obvious that Theorem 1.2.2 can be considered as a particular case of Theorem 1.2.1. Indeed relation (1.2.5) is a particular case of relation 0 (1.2.4), where q ." / .s/ is the characteristic function of the interval.

1.2.6 Renewal limits We define, for "  0, R1 x

."/

.1/ D

0

q ."/ .s/m.ds/ m."/ 1

:

(1.2.35)

Let us assume the following two conditions: D3 : F ."/ .s/ is a non-arithmetic distribution function for all " small enough, say "  "1 ; and F3 : q ."/ .s/ is directly Riemann integrable on Œ0; 1/ for all " small enough, say "  "2 . Note that, under condition F2 , it is enough to assume that q ."/ .s/ is a continuous function almost everywhere with respect to the Lebesgue measure on Œ0; 1/. In this case, condition F3 holds and the Lebesgue integration can be replaced with the Riemann integration in the right-hand side of the formula that defines x ."/ .1/. ."/ Condition M2 implies that m1 < 1 for all " small enough, say "  "3 .

1.2 Renewal theorem for perturbed renewal equation

41

Define "0 D min."1 ; "2 ; "3 /:

(1.2.36)

The renewal theorem implies the following statement. Theorem 1.2.3. Let conditions D3 , M2 , and F3 hold. Then, for "  "0 , x ."/ .t / ! x ."/ .1/ as t ! 1:

(1.2.37)

It should be noted that, under condition F2 , the expression for x ."/ .1/ given by formula (1.2.35) is finite for all " small enough without the assumption that q ."/ .s/ is a continuous function almost everywhere with respect to the Lebesgue measure on Œ0; 1/. This is true, since the formula involves the Lebesgue integration. Moreover, it can be proved that in this case the mean value of the x ."/ ./ over the interval Œ0; t  converges to x ."/ .1/ as t ! 1.

1.2.7 Convergence of renewal limits The following theorem supplements the asymptotic relations in (1.2.4) and (1.2.37). Theorem 1.2.4. Let conditions M2 , and F2 hold. Then x ."/ .1/ ! x .0/ .1/ as " ! 0:

(1.2.38)

.0/ Proof. According to condition M2 , we have m."/ 1 ! m1 2 .0; 1/ as " ! 0. So, it is sufficient to prove that condition F2 implies the following relation: Z 1 Z 1 ."/ q .s/m.ds/ ! q .0/ .s/m.ds/ < 1 as " ! 0: (1.2.39) 0

0

Conditions F2 .a/; .b/ imply that the function q ."/ .t / is bounded on every finite interval for all " small enough and that, by the Lebesgue theorem or Lemma 1.2.2, for every 0 < T < 1, Z T Z T q ."/ .s/m.ds/ ! q .0/ .s/m.ds/ as " ! 0: (1.2.40) 0

0

Also condition F2 .c/ implies that, for h in this condition, Z 1 lim lim jq ."/ .s/jm.ds/ T !1 0"!0 T

 lim

lim h

T !1 0"!0

X

sup

jq ."/ .t /j D 0:

(1.2.41)

r T = h r ht.rC1/h

Relations (1.2.40) and (1.2.41) imply in an obvious way relation (1.2.38), which proves Theorem 1.2.4.

42

1

Perturbed renewal equation

Remark 1.2.5. It follows from the proof that conditions M2 and F2 can be replaced with the following conditions: R1 ."/ .0/ M4 : m1 D 0 sF ."/ .ds/ ! m1 2 .0; 1/ as " ! 0; and F4 : q ."/ D

R1 0

q ."/ .s/m.ds/ ! q .0/ < 1 as " ! 0.

R1 .0/ The limiting constants may or may not be of the form m1 D 0 sF .0/ .ds/ and R1 q .0/ D 0 q .0/ .s/m.ds/. In any case, under conditions M4 and F4 , the following relation obviously holds: as " ! 0: x ."/ .1/ ! x .0/ .1/ D q .0/ =m.0/ 1

(1.2.42)

1.2.8 Perturbed renewal processes Here we give several statements that generalise some useful theorems of renewal theory to the model of a perturbed renewal equation. These theorems will be used in the next section, where Theorems 1.2.1 and 1.2.2 will be generalised to the model of a perturbed improper renewal equation. ."/ Let k , k D 1; 2; : : :, be a sequence of i.i.d. (independent identically distributed) random variables with a distribution function F ."/ .s/ the support of which is not conPn ."/ ."/ centrated in zero. The random variables n D kD1 k , n D 0; 1; : : :, can be interpreted as renewal moments and the corresponding renewal process can be defined as  ."/ .t / D max.n W n."/  t /, t  0. By definition,  ."/ .t / is the number of renewals in the interval Œ0; t . Let also t ."/ be a non-random function of " > 0 such that 0  t ."/ ! 1 as " ! 0. As is known (see, for example, Lo`eve (1955)), the conditions D4 : lim"!0 t ."/ .1  F ."/ .ht ."/ // D 0 for any h > 0; and M5 : lim"!0

R ht ."/ 0

xF ."/ .dx/ D m1 < 1 for any h > 0;

are necessary and sufficient for the following relation to hold: ."/

Œt  1 X

t ."/

."/

P

k1 ! m1 as " ! 0:

(1.2.43)

kD1

Lemma 1.2.3. Let conditions D4 and M5 hold. Then  ."/ .t ."/ / P 1 ! m1 as " ! 0: t ."/

(1.2.44)

1.2 Renewal theorem for perturbed renewal equation

43

Proof. According to the definition of the process  ."/ .t / for every t; x  0, Pf

."/

.t / > txg D P

 ŒtxC1 X

k."/

 t :

(1.2.45)

kD1

Relation (1.2.44) follows in obvious way from (1.2.43) and (1.2.45). Consider condition D4 together with the following condition: D5 : there exist g > 0 and G < 1 such that lim"!0 F ."/ .g/  G. Obviously conditions D4 and M5 together with condition D5 imply that the limiting constant satisfies m1 > 0. R ht ."/ xF ."/ .dx/  g.1  F ."/ .g/ and, therefore, Indeed, 0 Z lim

"!0 0

ht ."/

xF ."/ .dx/  g  g lim F ."/ .g/  g.1  G/ > 0: "!0

Remark 1.2.6. Condition D2 implies condition D5 . Also, conditions D2 and M2 imply that conditions D4 and M5 hold for any 0  t ."/ ! 1 as " ! 0. In this case, the .0/ .s/. constant m1 D m.0/ 1 is the mean of the limiting distribution function F

1.2.9 Convergence of moments for perturbed renewal processes In this section we give conditions for convergence of moments for perturbed renewal processes. Theorem 1.2.5. Let conditions D2 and M2 hold. Then for any 0  t ."/ ! 1 as " ! 0 and every r  1, !r  ."/ .t ."/ / .0/ ! .m1 /r as " ! 0: (1.2.46) E t ."/ Proof. Condition D2 implies condition D5 . Now, it follows from Lemma 1.1.1 and Remark 1.1.1 that for every r  1, !r  ."/ .t ."/ / lim E < 1: (1.2.47) "!0 t ."/ Now, by (1.2.44) and (1.2.47), the theorem follows from an appropriate variant of the Lebesgue theorem. In the case r D 1, relation (1.2.46) can be rewritten in an equivalent form, which is actually a triangular array analogue of the elementary renewal theorem.

44

1

Perturbed renewal equation

Theorem 1.2.6. Let conditions D2 and M2 hold. Then for any 0  t ."/ ! 1 as " ! 0, U ."/ .t ."/ / 1 ! .m.0/ as " ! 0: 1 / t ."/

(1.2.48)

Let us again consider the random variables  ."/ .t / which is the time that are times between the moment t and the first after t renewal moment defined by



."/

.t / D

 ."/X .t/C1

k."/  t;

t  0:

kD1

As is known, the probabilities Pf ."/ .t / > ug, as functions of t , satisfy the renewal equation Pf ."/.t / > ug D 1  F ."/ .t C u/ Z t C Pf ."/ .t  s/ > ugF ."/ .ds/; 0

Denote G ."/ .u/ D

Z

1 m."/ 1

u 0

.1  F ."/ .t // dt;

t  0:

(1.2.49)

u  0:

Theorem 1.2.7. Let conditions D2 and M2 hold. Then for any 0  t ."/ ! 1 as " ! 0 and u  0, Z 1 1 Pf ."/ .t ."/ / > ug ! .0/ .1  F .0/ .t C u// dt m1 0 Z 1 1 D .0/ .1  F .0/ .t // dt m1 u D 1  G .0/ .u/ as " ! 0:

(1.2.50)

Proof. In order to apply Theorem 1.2.1 to the renewal equation (1.2.49) we have to check that condition F2 holds for the forcing functions q ."/ .t / D 1  F ."/ .t C u/. Condition D2 implies that the functions q ."/ .t / D 1F ."/ .t Cu/ converge to q .0/ .t / D 1  F .0/ .t C u/, as " ! 1, at any continuity point t of q .0/ .t /. Since the functions q ."/ .t / are monotone, this convergence is easily seen to be locally uniform at every continuity point of q .0/ .t /. Since the set C0 of discontinuity points of the function q .0/ .t / D 1  F .0/ .t C u/ is at most countable m.C 0 / D 0. So, condition F2 .a/ holds. The functions q ."/ .t / D 1  F ."/ .t C u/ are bounded by 1 uniformly in " and

1.2 Renewal theorem for perturbed renewal equation

45

t . Hence, condition F2 .b/ also holds. Finally, using condition M2 , Remark 1.2.1, and monotonicity of q ."/ .t / we get that for every h > 0, lim h

lim

T !1 0"!0

D lim

X

X

lim h

T !1 0"!0

Z  lim

jq ."/ .t /j

sup

r T = h r ht.rC1/h

.1  F ."/ .rh C u//

r T = h 1

lim

T !1 0"!0 T h

.1  F ."/ .t C u// dt D 0:

(1.2.51)

Therefore condition F2 .c/ holds too. The application of Theorem 1.2.1 to the renewal equation (1.2.49) completes the proof. d

By ."/ .t /, t 2 T ! .0/ .t /, t 2 T as " ! 0 we will denote convergence of stochastic processes in distribution, i.e., weak convergence of finite-dimensional distributions of the stochastic processes ."/ .t / to the corresponding finite-dimensional distributions of the stochastic process .0/ .t / on the set T as " ! 0. Also denote by Dr."/ the set of discontinuity points of the distribution function of P S ."/ ."/ is the random variables n."/ D nkD1 k."/ . Let also D ."/ D 1 rD0 Dr . The set D at most countable. We need the following auxiliary proposition. Lemma 1.2.4. Let condition D2 hold. Then d

 ."/ .t /; t 2 DN .0/ !  .0/ .t /; t 2 DN .0/ as " ! 0:

(1.2.52)

Proof. By the definition of the renewal process,  ."/ .t /, t  0, for any positive integer tn  0, rn D 0; 1; : : :, and n; m D 1; 2; : : :, we have Pf ."/ .tn /  rn ; n D 1; : : : ; mg D P

X rn

 ."/

k  tn ; n D 1; : : : ; m :

(1.2.53)

kD1

Relation (1.2.52) easily follows from (1.2.53), the independence of the random vari."/ ables k , k D 1; 2; : : :, and condition D2 . Lemma 1.2.5. If conditions D2 and M2 hold, then for any points tn 2 DN .0/ and n; m  1, E

m Y nD1



."/

.tn / ! E

m Y nD1

 .0/ .tn / as " ! 0:

(1.2.54)

46

1

Perturbed renewal equation

Proof. Since conditions D2 and M2 imply condition D5 , it follows from Lemma 1.1.1 that there exists a constant K < 1 such that lim E ."/ .t /r  Kt r ;

t  0;

"!0

r D 1; 2; : : : :

(1.2.55)

Relation (1.2.54) follows at once from (1.2.55) and Lemma 1.2.4. The following relation can be obtained from (1.1.10) by summing over r  n: Pf ."/ .t C h/   ."/ .t /  ng D ı.0; n/Pf ."/ .t / > hg Z h C Pf ."/ .h  s/ C 1  ngPf ."/.t / 2 dsg; n D 0; 1; : : : : 0

(1.2.56)

Theorem 1.2.8. Let condition D2 and M2 hold. Then, for any 0  t ."/ ! 1 as " ! 0 and h  0 and n D 0; 1; : : :, the following relation holds: Pf ."/ .t ."/ C h/   ."/ .t ."/ /  ng ! ı.0; n/.1  G .0/ .h// C D1

.0/ Hh .n/

Z

h 0

Pf .0/ .h  s/ C 1  ngG .0/ .ds/

as " ! 0:

(1.2.57)

Proof. We would like to apply Lemma 1.2.2 to the integrals in the left-hand side of (1.2.56). By Lemma 1.2.4 we have at every point of continuity of the limiting function f .0/ .s/, f ."/ .s/ D Pf ."/ .h  s/ C 1  ngŒ0;h .s/ ! f .0/ .s/ D Pf .0/ .h  s/ C 1  ngŒ0;h .s/ as " ! 0:

(1.2.58)

Functions f ."/ .s/ are monotone on the interval Œ0; h. Hence, this convergence is easily seen to be locally uniform in each point of continuity of the limiting function f .0/ .s/. The set D of points of discontinuity of the function f .0/ .s/ is at most countable. Denote ."/ .A/ D Pf ."/ .t / 2 A \ Œ0; hg. Weak convergence of these measures is implied by Theorem 1.2.7. Moreover, ."/ .R1 / D Pf ."/ .t ."/ /  hg ! 1G .0/ .h/ as " ! 0, since G .0/ .h/ is a distribution function continuous in h. Finally, the limiting measure .0/ .A/ is absolutely continuous on the interval Œ0; h, according to the formula (1.2.50). So, .0/ .D/ D 0. Thus, all conditions of Lemma 1.2.2 are satisfied. Applying this lemma in (1.2.56) and using relation (1.2.50) from Theorem 1.2.7 we get relation (1.2.57).

47

1.2 Renewal theorem for perturbed renewal equation

1.2.10 Stationary renewal processes .0/

Let Qk , k D 1; 2; : : :, be a sequence of nonnegative independent random variables .0/

such that the random variables Q1 has a distribution function G .0/ .t /, while the ran.0/ dom variables Qk , k > 1, are identically distributed with a distribution function P .0/ .0/ F .0/ .t /. Denote Q k D nkD1 Qk , n D 0; 1; : : :, and introduce a renewal process .0/

constructed from the random variables Q k , Q .0/ .t / D max.n W Q n.0/  t /;

t  0:

(1.2.59)

As is known (see, for example, Feller (1971)) the renewal process Q .0/ .t /, t  0, is a stationary renewal process (process with stationary increments). In particular, the distribution of the increments Q .0/ .t Ch/ Q .0/ .t / of this process depends on h  0 but not on t  0. They are given by expression in the right-hand side of relation (1.2.57), i.e., for all h; t  0 and n D 0; 1; : : :, PfQ .0/ .t C h/  Q .0/ .t /  ng D PfQ .0/ .h/  ng D 1  Hh.0/ .n/:

(1.2.60)

Since the renewal process Q .0/ .t / is stationary, E.Q .0/ .t C h/  Q .0/ .t // D EQ .0/ .h/ X .0/ .0/ D .1  Hh .n// D h=m1 ;

h; t  0:

(1.2.61)

n1

In principle, it would be possible to show that under conditions of Theorem 1.2.8 we have the following more general relation: d

 ."/ .t ."/ C h/   ."/ .t ."/ /; h  0 ! Q .0/ .h/; h  0 as " ! 0:

(1.2.62)

1.2.11 Convergence of moments for increments of perturbed renewal processes We now return to relation (1.1.11), which has the following form: E.

."/

.t C h/  

."/

r

Z

.t // D

h 0

E. ."/ .h/ C 1/r Pf ."/ .t / 2 dsg;

r  1: (1.2.63)

Theorem 1.2.9. Let conditions D2 and M2 hold. Then for any 0  t ."/ ! 1 as " ! 0 and h  0 and r D 1; 2; : : :, the following relation holds: E.

."/

.t

."/

C h/  

."/

.t

."/

r

// !

Z

h 0

E. .0/ .h/ C 1/r G .0/ .ds/ as " ! 0: (1.2.64)

48

1

Perturbed renewal equation

Proof. The proof is analogous to the proof of Theorem 1.2.8. Lemma 1.2.2 can be applied to the integrals in the left-hand side of (1.2.63). In this case, the functions Pf ."/ .h  s/ C 1  ngŒ0;h .s/ have to be replaced with f ."/ .s/ D E. ."/ .h  s/ C 1/r Œ0;h .s/ but we need to use the same measures ."/ .A/ D P f ."/.t / 2 A \ Œ0; hg. By Lemma 1.2.5 we have at every point of continuity of the limiting function f .0/ .s/, f ."/ .s/ D E. ."/ .h  s/ C 1/r Œ0;h .s/ ! f .0/ .s/ D E. .0/ .h  s/ C 1/r Œ0;h .s/ as " ! 0:

(1.2.65)

Functions f ."/ .s/ are monotone on the interval Œ0; h, and f .0/ .h/ D E. .0/ .h/ C 1/ < 1. Hence, this convergence is easily seen to be locally uniform in each point of continuity of the limiting function f .0/ .s/, i.e., at every point s 2 D, where D is the same set as in the proof of Theorem 1.2.8. A further reasoning is absolutely analogous to the one given in the proof of Theorem 1.2.8. One should observe that the proof could also be completed by applying a corresponding variant of the Lebesgue theorem. Theorem 1.2.8 means that r

d

 ."/ .t ."/ C h/   ."/ .t ."/ / ! Q .0/ .h/ as " ! 0;

(1.2.66)

where Q .0/ .h/ is a discrete random variable such that PfQ .0/ .h/  ng D 1  Hh.0/ .n/. At the same time Lemma 1.1.2 imply that there exist constants KŒh; r < 1 such that lim E. ."/ .t ."/ C h/   ."/ .t ."/ //r  KŒh; r:

"!0

(1.2.67)

Relations (1.2.66) and (1.2.67) imply (1.2.64). Note also that the limit expression in the right-hand side of (1.2.64) in Theorem 1.2.9 coincides with EQ .0/ .h/r , and relation (1.2.64) could be rewritten in the following form, for h  0, r D 1; 2; : : :, E. ."/ .t ."/ C h/   ."/ .t ."/ //r ! EQ .0/ .h/r as " ! 0:

(1.2.68)

Theorem 1.2.9 is in a sense a generalisation of Theorem 1.2.2. For the case r D 1 the assertions of both theorems coincide in view of the relation U.t / D E ."/ .t / C 1.

1.3 Improper perturbed renewal equation In this section we generalise the renewal Theorem 1.2.1 to a model in which the distribution functions F ."/ .t / that generate the renewal equation (1.2.1) can be improper for " > 0 but proper for " D 0, i.e., the total variation satisfies F ."/ .1/  1 for " > 0 but F .0/ .1/ D 1. In such a case we say that (1.2.1) is an improper renewal equation.

1.3 Improper perturbed renewal equation

49

1.3.1 Renewal theorem for improper perturbed renewal equation All general facts and definitions concerning the renewal equation are translated to the case of an improper renewal equation without any changes. In particular, the renewal function is defined by the same formula and a unique solution of the renewal equation is defined via the renewal function by the same formula as in the case of the proper renewal equation. We define the first moment of a distribution F ."/ .s/ as usual by the formula Z 1 ."/ sF ."/ .ds/: m1 D 0

This functional can be finite also in the case where the total variation F ."/ .1/ < 1. ."/ However, in this case, m1 can not be interpreted as the expectation of an improper random variable with the distribution F ."/ .s/. Any such random variable has an infinite expectation since it takes the value C1 with the positive probability 1F ."/ .1/. Conditions D2 and M2 take in this case the following form: D6 : F ."/ ./ ) F .0/ ./ as " ! 0, where F .0/ .s/ is a proper non-arithmetic distribution function; and ."/

.0/

M6 : m1 ! m1 < 1 as " ! 0. Condition D6 means that F ."/ .t / ! F .0/ .t / as " ! 0 in every point t that is a point of continuity for the limiting distribution function. This condition also includes the assumption F .0/ .1/ D 1. Thus, D6 also implies the following relation: F ."/ .1/ ! F .0/ .1/ D 1 as " ! 0:

(1.3.1)

Let f ."/ D 1  F ."/ .1/ denote the defect of the distribution F ."/ .t /. In this case, we also consider a new type of condition that would balance the rate of growth of time t ."/ , which is supposed to be a non-random nonnegative function of ", and the rate of deficiency of the renewal equation determined by the defect f ."/ D 1  F ."/ .1/, B1 : 0  t ."/ ! 1 and f ."/ ! 0 as " ! 0 in such a way that f ."/ t ."/ ! , where 0    1. The main result of this section is the following theorem that generalises the classical renewal theorem to the model of improper perturbed renewal equations. Theorem 1.3.1. Let conditions D6 , M6 , F2 hold, and 0  t ."/ ! 1 as " ! 0 such that condition B1 holds. Then R 1 .0/ q .s/ ds ."/ ."/ =m.0/ 1 x .t / ! e  0 as " ! 0: (1.3.2) .0/ m1

50

1

Perturbed renewal equation

Theorem 1.3.1 differs from Theorem 1.2.1. There appears a new exponential factor in the left-hand side of the asymptotic relation (1.3.2). It describes the asymptotical behaviour of a solution of the renewal equation as the time t ."/ tends to infinity. The appearance of this factor is caused by the deficiency of the distribution functions F ."/ .s/ that generate the renewal equation (1.2.1). At the same time, Theorem 1.3.1 assumes some restriction on the rate of growth of time t ."/ in condition B1 , while Theorem 1.2.1 imposes no restrictions on the rate at which t ."/ tends to infinity. A proper renewal equation is obtained if f ."/  0. In this case, condition B1 automatically holds, since f ."/ t ."/  0 and  D 0. Also note that in this case, t ."/ can tend to infinity in an arbitrary way. The exponential factor in the left-hand side of (1.3.2) takes the value 1. Theorem 1.3.1 is reduced to Theorem 1.2.1. Also note that we admit the case where  D C1. In this case, the exponential factor in the left-hand side of (1.3.2) takes the value 0 and relation (1.3.2) takes the form x ."/ .t ."/ / ! 0 as " ! 0. As in the case of a proper renewal equation, the main part of the proof of Theorem 1.3.1 is based on a generalisation of Blackwel’s theorem to the model of an improper perturbed renewal equation. Theorem 1.3.2. Let conditions D6 , M6 hold, and 0  t ."/ ! 1 as " ! 0 such that condition B1 holds. Then for any h  0, .0/

as " ! 0: U ."/ .t ."/ C h/  U ."/ .t ."/ / ! e =m1  h=m.0/ 1

(1.3.3)

1.3.2 Perturbed renewal processes with improper inter-renewal times We postpone the proof of Theorems 1.3.1 and 1.3.2. We first formulate and prove analogues of Theorems 1.2.8 and 1.2.9 for the model of perturbed improper renewal processes. These theorems will be used in the proofs of Theorems 1.3.1 and 1.3.2. ."/ Let k , k D 1; 2; : : :, be improper i.i.d. random variables with values in the interval Œ0; 1 and a distribution F ."/ .t /. This means that these random variables are improper ."/ and take value C1 with the probability f ."/ D 1  F ."/ .1/. We also define n D Pn ."/ kD1 k , n D 0; 1; 2; : : :, and construct the corresponding renewal process,  ."/ .t / D max.n W n."/  t /;

t  0:

Let us define proper distribution functions concentrated on Œ0; 1/, FN ."/ .t / D F ."/ .t /=F ."/ .1/;

t  0:

."/

Let also Mk , k D 1; 2; : : :, be proper i.i.d. random variables with values in the interP ."/ ."/ val Œ0; 1/ and distribution FN ."/ .t /. We also define M n D nkD1 Mk ; n D 0; 1; 2; : : :, and construct the corresponding renewal process, M ."/ .t / D max.n W n."/  t /;

t  0:

51

1.3 Improper perturbed renewal equation

Finally, let M ."/ be a random variable which (a) is defined on the same probability space as the random variables Mk."/ , k D 1; 2; : : :; and (b) independent of these random variables; (c) has a geometric distribution with the parameter f ."/ , i.e., PfM ."/ D ng D f ."/ .1  f ."/ /n1 ;

n D 1; 2; : : : :

The following formula connecting the renewal processes  ."/ .t / and M ."/ .t / plays a crucial role in the sequel. It states that for any 0  t1  t2      tn < 1, integer 0  r1  r2      rn and n  1, Pf ."/ .tk /  rk ; k D 1; : : : ; ng  X rk ."/

l  tk ; k D 1; : : : ; n DP lD1 ."/

< 1; l D 1; : : : ; rn g X  rk ."/ ."/ P

l  tk ; k D 1; : : : ; n= l < 1; l D 1; : : : ; rn

D Pf l

lD1

DF

."/

rn

.C1/

P

X rk

."/

Ml

  tk ; k D 1; : : : ; n

lD1

D PfM ."/  rn g  PfM ."/ .tk /  rk ; k D 1; : : : ; ng D Pfmin.M ."/ ; M ."/ .tk //  rk ; k D 1; : : : ; ng;

(1.3.4)

where the random variable M ."/ and the renewal process M ."/ .t /, t  0, are independent of the right-hand side of (1.3.4). Denote by  an exponentially distributed random variable with parameter 0 <  < 1, i.e., Pf > t g D e t , t  0. We also denote by 0 an improper random variable that takes value C1 with probability 1 and by 1 a random variable that takes value 0 with probability 1. Lemma 1.3.1. Let conditions D6 , M6 hold, and 0  t ."/ ! 1 as " ! 0 such that condition B1 holds. Then d

.0/

 ."/ .t ."/ /=t ."/ ! min. ; 1=1 / as " ! 0:

(1.3.5)

Theorem 1.3.3. Let conditions D6 , M6 hold, and 0  t ."/ ! 1 as " ! 0 such that condition B1 holds. Then, for all r D 1; 2; : : :, the following relation holds: r E. ."/ .t ."/ /=t ."/ /r ! E.min. ; 1=.0/ 1 // as " ! 0:

(1.3.6)

52

1

Perturbed renewal equation

Theorem 1.3.4. Let conditions D6 , M6 hold, and 0  t ."/ ! 1 as " ! 0 such that condition B1 holds. Then .0/

U ."/ .t ."/ /=t ."/ ! E min. ; 1=1 / D

.0/ 1 .1  e =1 / as " ! 0: 

(1.3.7)

Proof of Lemma 1.3.1 and Theorems 1.3.3 and 1.3.4. It is obvious that under conditions B1 , d

M ."/ =t ."/ !  as " ! 0:

(1.3.8)

Also, conditions D6 and B1 imply that condition D2 holds for the distribution func.0/ tions FN ."/ .t / with the same limiting mean 1 . It then follows from Lemma 1.2.3 that P

.0/

M ."/ .t ."/ /=t ."/ ! 1=1 d

as " ! 0:

(1.3.9)

P

As is known, if ."/ ! .0/ as " ! 0 and ."/ ! .0/ as " ! 0, where .0/ D d

const with probability 1, then random vectors . ."/ ; ."/ / ! . .0/ ; .0/ / as " ! 0. d

Moreover, for any continuous function g.x; y/, random variables g. ."/ ; ."/ / ! g. .0/ ; .0/ / as " ! 0. Using this remark and taking into consideration the representation (1.3.4) we get from (1.3.8) and (1.3.9) that d

.0/

 ."/ .t ."/ /=t ."/ ! min. ; 1=1 / as " ! 0:

(1.3.10)

Since condition D2 holds for the distribution functions FN ."/ .t /, condition D4 also holds, and it follows from Lemma 1.1.1 and Remark 1.1.1 that, for every r  1, lim E. ."/ .t ."/ /=t ."/ /r  lim E.M ."/ .t ."/ /=t ."/ /r < 1:

"!0

"!0

(1.3.11)

Relation (1.3.6) easily follows from (1.3.10) and (1.3.11). As far as relation (1.3.7) is concerned, it is a particular case of (1.3.6), since U ."/ .t / D E ."/ .t / C 1. Now, we formulate and prove analogues of Theorems 1.2.8 and 1.2.9. Theorem 1.3.5. Let conditions D6 , M6 hold, and 0  t ."/ ! 1 as " ! 0 such that condition B1 holds. Then for all h  0 and n D 1; 2; : : :, the following relation holds: .0/

.0/

Pf ."/ .t ."/ C h/   ."/ .t ."/ /  ng ! e =m1 .1  Hh .n// as " ! 0:

(1.3.12)

1.3 Improper perturbed renewal equation

53

Proof. It follows from representation (1.3.7) that for any t; h  0 and n D 1; 2; : : :, Pf ."/ .t C h/   ."/ .t /  ng D Pfmin.M ."/ ; M ."/ .t C h//  min.M ."/ ; M ."/ .t //  ng;

(1.3.13)

and hence for any t; h  0 and n D 1; 2; : : :, Pf ."/ .t C h/   ."/ .t /  ng D PfM ."/ > M ."/ .t C h/; M ."/ .t C h/  M ."/ .t /  ng C PfM ."/ .t C h/  M ."/ > M ."/ .t /; M ."/  M ."/ .t /  ng:

(1.3.14)

It is obvious that condition B1 holds for the functions t ."/ C h if it holds for the functions t ."/ . Hence, similarly to (1.3.9) we have that, for any h  0, P

.0/

M ."/ .t ."/ C h/=t ."/ ! 1=1

as " ! 0:

(1.3.15)

From (1.3.15) and (1.3.8) and (1.3.9) it follows that for any h  0, .M ."/ =t ."/ ; M ."/ .t ."/ /=t ."/ ; M ."/ .t ."/ C h/=t ."/ / d

.0/ ! . ; 1=m.0/ 1 ; 1=m1 / as " ! 0:

(1.3.16)

Now, (1.3.16) yields for any h  0, PfM ."/ .t ."/ C h/  M ."/ > M ."/ .t ."/ /g D PfM ."/ .t ."/ C h/=t ."/  M ."/ =t ."/ > M ."/ .t ."/ /=t ."/ g ! 0 as " ! 0:

(1.3.17)

Theorem 1.2.8 can be applied to the processes M ."/ .t ."/ /, since conditions D6 , M6 and B1 imply that conditions D2 and M2 hold for the distribution functions FN ."/ .t /. Since M ."/ and M ."/ .t ."/ C h/  M ."/ .t ."/ / are independent, we get using this remark and (1.3.8) that, for any h  0, d

.M ."/ =t ."/ ; M ."/ .t ."/ C h/  M ."/ .t ."/ // ! . ; Q .0/ .h// as " ! 0;

(1.3.18)

where the random variables  and Q .0/ .h/ are independent and the random variable Q .0/ .h/ is defined in Subsection 1.2.9. According to the definition of the random variables Q .0/ .h/, .0/

PfQ .0/ .h/  ng D 1  Hh .n/;

n D 1; 2; : : : :

54

1

Perturbed renewal equation

It follows from (1.3.18) and (1.3.15) that for any h  0, .M ."/ =t ."/  M ."/ .t ."/ C h/=t ."/ ; M ."/ .t ."/ C h/  M ."/ .t ."/ // d

! .  1=m.0/ Q .0/ .h// as " ! 0; 1 ;

(1.3.19)

where the random variables  and Q .0/ .h/ are independent. .0/ Since the distribution function of   1=m1 is continuous and Q .0/ .h/ has a discrete distribution function, from (1.3.19) we obtain for any h  0 and n D 1; 2; : : :, PfM ."/  M ."/ .t ."/ C h/ > 0; M ."/ .t ."/ C h/  M ."/ .t ."/ /  ng .0/

.0/

! Pf  1=m1 > 0g.1  Hh .n// .0/

.0/

D expf=m1 g.1  Hh .n// as " ! 0:

(1.3.20)

Relation (1.3.12) follows from (1.3.14), (1.3.17) and (1.3.20). Define the random variables h D Q .0/ .h/.  1=m.0/ 1 /; where  and Q .0/ .h/ are independent random variables, and Q .0/ .h/ is the number of renewals in the stationary renewal process defined in Subsection 1.2.10. By definition, ( .0/ .0/ 0 with probability 1  expf=m1 gŒ1  Hh .1/; h D .0/ .0/ n with probability expf=m.0/ n  1: 1 g.Hh .n/  Hh .n C 1//; It also follows from the definition of the random variables h that .0/

Ehr D expf=m1 gE.Q .0/ .h//r ;

r D 1; 2; : : : ;

(1.3.21)

and, in particular, .0/

.0/

Eh D expf=m1 gh=m1 :

(1.3.22)

Theorem 1.3.6. Let conditions D6 , M6 hold, and 0  t ."/ ! 1 as " ! 0 such that condition B1 holds. Then for all h  0 and r D 1; 2; : : :, the following relation holds: .0/

E. ."/ .t ."/ C h/   ."/ .t ."/ /r ! e =m1 E.Q .0/ .h//r as " ! 0:

(1.3.23)

Proof. Obviously for all t; h  0,  ."/ .t C h/   ."/ .t / D min.M ."/ ; M ."/ .t C h//  min.M ."/ ; M ."/ .t //  M ."/ .t C h/  M ."/ .t /:

(1.3.24)

1.3 Improper perturbed renewal equation

55

Theorem 1.2.9 can be applied to the processes M ."/ .t ."/ /, since D6 and B1 imply that condition D2 holds. Using (1.3.24) and the above we have for all t; h  0 and r D 1; 2; : : : that lim E. ."/ .t C h/   ."/ .t //r  lim E.M ."/ .t C h/  M ."/ .t //r < 1:

"!0

"!0

(1.3.25)

Relation (1.3.23) follows in an obvious way from relation (1.3.12) given in Theorem 1.3.5 and from (1.3.25).

1.3.3 Proof of Theorems 1.3.1 and 1.3.2 The renewal function generated by the distribution function F ."/ .t / is connected with the corresponding renewal function via the same relation as in the model of a proper renewal equation, U ."/ .t / D E ."/ .t / C 1;

t  0:

(1.3.26)

Using relation (1.3.26) and formula (1.3.22) we can conclude that Theorem 1.3.2 is a particular case of Theorem 1.3.6 if r D 1. A unique solution of the renewal equation is given by the formulas Z

x ."/ .t / D

Z D

t 0 t 0

q ."/ .t  s/U ."/ .ds/ q ."/ .v/U ."/ .t  dv/;

t  0:

(1.3.27)

The proof of Theorem 1.3.1 is same as that given for Theorem 1.2.1 in the previous section. Theorem 1.3.2 and condition F2 permit to apply Lemma 1.2.2 to the functions q ."/ .v/ and the measures ."/ .A/ D U ."/ .t ."/  A \ Œ0; T / for any T < 0, which gives the following relation: Z 0

T

q ."/ .v/U ."/ .t ."/  dv/ !

expfc=m.0/ 1 g .0/ m1

Z

T 0

q .0/ .s/m.dv/ as " ! 0: (1.3.28)

Denote by UN ."/ .t / the renewal function generated by the proper distribution function FN ."/ .t /. It follows from (1.3.24) and (1.3.26) that for any t; h  0, U ."/ .t C h/  U ."/ .t /  UN ."/ .t C h/  UN ."/ .t /

(1.3.29)

and, therefore, Z

t ."/ T

jq

."/

.v/jU

."/

.t

."/

Z  dv/ 

t ."/

T

jq ."/ .v/jUN ."/ .t ."/  dv/:

(1.3.30)

56

1

Perturbed renewal equation

Since conditions D6 , M6 and B1 imply that conditions D2 and M2 hold for the distribution functions FN ."/ .t /, we can apply Lemma 1.1.3 to the renewal functions UN ."/ .t / and get, for any interval I D Œa; b of length h D b  a, that 0 lim sup UN ." / .t C I /  K1 C K2 h < 1;

"!0 t0

(1.3.31)

where K1 ; K2 < 1. Using (1.3.30), (1.3.31), and condition F2 we find for h > 0 that Z lim

t ."/

"!0 T

jq ."/ .v/jU ."/ .t ."/  dv/ Z

 lim

t ."/

"!0 T

 lim

jq ."/ .v/jUN ."/ .t ."/  dv/

X

"!0

jq ."/ .t /j

sup

r ŒT = h r ht.rC1/h 0

0

0

0

 .UN ." / .t ." /  rh/  UN ." / .t ." /  .r C 1/h// X K1 C K2 h 0   lim h sup jq ." / .t /j ! 0 as T ! 0: "!0 h T h r ht.rC1/h r

(1.3.32)

h

From relations (1.3.28) and (1.3.32) it follows that lim x

"!0

."/

.t

."/

Z / D lim

t ."

0/

"!0 0

D lim lim

q ."/ .t ."/  s/U ."/ .ds/ Z

T

T !1 "!0 0

D lim

expfc=m.0/ 1 g

T !1

D

q ."/ .t ."/  s/U ."/ .ds/

.0/

m1

expfc=m.0/ 1 g .0/ m1

Z

T 0

Z

1 0

q .0/ .v/m.dv/

q .0/ .v/m.dv/:

(1.3.33)

This completes the proof.

1.4 Perturbed renewal equation under Cram´er type conditions In this section, we continue studies of the asymptotic behaviour of the solution x ."/ .t ."/ / of a perturbed renewal equation in the model with the distribution functions

1.4 Perturbed renewal equation under Cram´er type conditions

57

F ."/ .t / that generate the renewal equation (1.2.1) that may be improper for " > 0, i.e., F ."/ .1/  1 for " > 0. In the limiting case " D 0, we consider both cases when either F .0/ .1/ D 1 or F .0/ .1/ < 1. However, we additionally impose a Cram´er type condition on the distributions F ."/ .t /. As we shall see, this additional assumption allows to omit the restriction on the rate of growth of the time t ."/ imposed by the balancing condition B1 . We also consider a model in which the perturbed renewal equations are generated by some finite measures F ."/ .dt / that weakly converge to some limiting finite measure F .0/ .dt / with the total variation less, equal or greater than 1.

1.4.1 Perturbed renewal equations generated by asymptotically proper distribution functions We consider the case when condition D6 holds, i.e., F ."/ ./ ) F .0/ ./ as " ! 0, where F .0/ .s/ is a proper non-arithmetic distribution function. As was mentioned in Subsection 1.3.1, this condition also contains the assumption F .0/ .1/ D 1 that automatically implies that F ."/ .1/ ! F .0/ .1/ D 1 as " ! 0 or, which is the same, that f ."/ D 1  F ."/ .1/ ! f .0/ D 0 as " ! 0. Now, we assume the following Cram´er type condition: R1 C1 : There exists ı > 0 such that lim0"!0 0 e ıs F ."/ .ds/ < 1. Obviously, conditions D6 and C1 imply condition M6 , i.e., .0/ m."/ 1 ! m1 < 1 as " ! 0:

Let us introduce the moment generating function, Z 1  ."/ ./ D e s F ."/ .ds/; 0

(1.4.1)

  0:

By the definition,  ."/ ./ is a nonnegative and nondecreasing function. From condition C1 we have that  ."/ .ı/ < 1 for " small enough, say "  "1 . The following characteristic equation plays an important role in what follows:  ."/ ./ D 1:

(1.4.2)

Lemma 1.4.1. Let conditions D6 and C1 hold. Then, for any ˇ 2 .0; ı, there exists "0 D "0 .ˇ/ > 0 (defined below in formula (1.4.4)) such that, for every 0  "  "0 : .˛/  ."/ .ˇ/ 2 .1; 1/; .ˇ/ there exists a unique nonnegative root ."/ of equation (1.4.2); ./ ."/ < ˇ; .ı/ ."/ ! .0/ D 0 as " ! 0. Proof. By condition D6 , the distribution function F ."/ .s/ is not concentrated in the origin, i.e., F ."/ .0/ < 1 for " small enough, say "  "2 . It is obvious that, for "  min."1 ; "2 /, the function " ./ is nonnegative, strongly increasing, and continuous on the interval Œ0; ı. Note also that  ."/ .0/ D F ."/ .1/  1.

58

1

Perturbed renewal equation

Since the distribution function F .0/ .s/ is proper and is not concentrated in 0, equation (1.4.2) for " D 0 has the unique root .0/ D 0, i.e.,  .0/ .0/ D 1, and  .0/ ./ > 1 for any  2 .0; ı. Now, using condition D6 , we have Z T lim " .ı/  lim lim e sı F" .ds/ "!0

T !1 "!1 0

Z

D lim

T !1 0

T

e sı F0 .ds/ D 0 .ı/ > 1:

(1.4.3)

It follows from (1.4.3) that, for any ˇ 2 .0; ı, we have " .ˇ/ > 1 for " small enough, say "  "3 .ˇ/. Let us now define "0 D "0 .ˇ/ D min."1 ; "2 ; "3 .ˇ//:

(1.4.4)

It follows from the remarks above that there exists a unique nonnegative root of equation (1.4.2) for "  "0 . This solution satisfies "  ˇ. Since ˇ can be any positive number less or equal ı, we also see that the relation " ! 0 holds as " ! 0. Remark 1.4.1. It is worth noting that the assumption that the limiting distribution function F0 .s/ is non-arithmetic was not used in the proof of Lemma 1.4.1. This assumption can be replaced in condition D6 with a weaker assumption, actually used in the proof, that the limiting distribution function F0 .s/ is not concentrated in zero. Now we assume the following condition: F5 :

(a) limu!0 lim0"!0 supjvju jq ."/ .t C v/  q .0/ .t /j D 0 almost everywhere with respect to the Lebesgue measure on Œ0; 1/; (b) lim0"!0 sup0tT jq ."/ .t /j < 1 for every T  0; P (c) limT !1 lim0"!0 h r T = h supr ht.rC1/h e t jq ."/ .t /j D 0 for some h > 0 and some  > 0.

Note that condition F5 can be formulated in the following equivalent form: F05 : There exists  > 0 such that the function e t q ."/ .t /, t  0 satisfies condition F2 . Theorem 1.4.1. Let conditions D6 , C1 , and F5 hold. Then, for any 0  t ."/ ! 1 as " ! 0, R 1 .0/ q .s/ ds x ."/ .t ."/ / as " ! 0: (1.4.5) ! 0 ."/ ."/ .0/ expf t g m1

59

1.4 Perturbed renewal equation under Cram´er type conditions

Proof. We use the exponential transformation of renewal equation (1.2.1). Denote FQ ."/ .t / D

Z

t 0

e

."/ t

F ."/ .ds/; qQ ."/ .t / D e 

."/ t

q ."/ .t /; xQ ."/ .t / D e 

."/ t

x ."/ .t /; t  0:

Note first of all that, since ."/ is a root of equation (1.4.2) by the definition, it implies that FQ ."/ .1/ D 1, i.e., FQ ."/ .t / is a proper distribution function. Since x ."/ .t / is a solution of the renewal equation (1.2.1), the function xQ ."/ .t / is a solution of the renewal equation xQ

."/

.t / D qQ

."/

Z .t / C

t 0

xQ ."/ .t  s/FQ ."/ .ds/;

t  0:

(1.4.6)

Since 0  ."/ ! 0, conditions D6 and C1 imply in an obvious way that the distribution functions FQ ."/ .t / satisfy conditions D2 and M2 . Moreover, for the corresponding limiting distribution, we have FQ .0/ .t /  F .0/ .t / and, therefore, .0/ m Q1

Z D

1 0

.0/ s FQ .0/ .ds/ D m1 D

Z

1 0

sF .0/ .ds/:

(1.4.7)

Also the relation 0  ."/ ! 0 and condition F5 imply that the functions qQ ."/ .t / satisfy condition F2 . Moreover, for the corresponding limiting function, qQ .0/ .t /  q .0/ .t / and, therefore, Z 0

1

qQ

.0/

Z .s/ ds D

1 0

q .0/ .s/ ds:

(1.4.8)

By applying Theorem (1.2.1) to equation (1.4.6) and taking the remarks made above and relations (1.4.7) and (1.4.8) into consideration we get the relation R1 xQ

."/

.t

."/

/De

."/ t ."/ ."/

x

.t

."/

/!

0

q .0/ .s/ ds .0/

as " ! 0;

(1.4.9)

m1 which is equivalent to the relation (1.4.5).

As was mentioned above, the main difference between Theorem 1.3.1 and Theorem 1.4.1 is that no balancing condition B1 is assumed in the Theorem 1.4.1. This condition restricts the rate of growth of the time parameter t ."/ to 1. However, to be able not to use this condition, the Cram´er type condition C1 should be imposed on the distribution functions F ."/ .t /. We use the symbol a."/  b ."/ as " ! 0 to denote that a."/ =b ."/ ! 1 as " ! 0. The following lemma provides additional information about asymptotics for the coefficients ."/ .

60

1

Perturbed renewal equation

Lemma 1.4.2. Let conditions D6 and C1 hold. Then, the following asymptotical relation holds: ."/  f ."/ =m."/ 1 as " ! 0:

(1.4.10)

Proof. It was proved in Lemma 1.4.1 that under conditions D6 and C1 there exists a unique nonnegative root ."/ of equation (1.4.2) for all " small enough, and that ."/ ! 0 as " ! 0. Let us use Taylor’s expansion for the function e t which can be given in the following form (taking onto consideration that this function and its derivatives are increasing) e s

."/

."/

D 1 C s."/ C s 2 .."/ /2 e s  ."/ .s/=2;

(1.4.11)

where  ."/ .s/; s  0, is a continuous function such that 0  " .s/  1;

s  0:

(1.4.12)

As was shown in the proof of Lemma 1.4.1, for any ˇ 2 .0; ı, there exists "0 .ˇ/ such that "  ˇ for "  "0 .ˇ/. By condition C1 , Z 1 Z 1 2 ˇs ."/ lim s e F .ds/  lim c2 e ıs F ."/ .ds/ < 1; (1.4.13) 0"!0 0

0"!0

0

where c2 D sups0 s 2 e .ıˇ /s < 1. It follows from relation .1:4:13/ that there exists "4 D "4 .ˇ/ > 0 such that for "  "4 , Z 1 1 s 2 e sˇ F ."/ .ds/ < 1: (1.4.14) M D sup 2 ""4 0 Define, "5 D "5 .ˇ/ D min."0 .ˇ/; "4 .ˇ//:

(1.4.15)

By integrating in (1.4.11) and using (1.4.12) and (1.4.14), we get for "  "5 that Z 1 ."/ ."/ e s F ."/ .ds/ D 1  f ."/ C m1 ."/ C .."/ /2 M  ."/ ; (1.4.16) 0

where R1 

."/

D

."/

s 2 e s  ."/ .s/F ."/ .ds/ R1 2 Œ0; 1: sup""4 0 s 2 e sˇ F ."/ .ds/ 0

(1.4.17)

1.4 Perturbed renewal equation under Cram´er type conditions

61

Therefore, equation (1.4.2) can be rewritten in the form ."/

m1 ."/ C .."/ /2 M  ."/ D f ."/ ;

(1.4.18)

or, equivalently, ."/

m1 ."/ .1 C ."/ M  ."/ =m."/ 1 / D 1: ."/ f ."/

(1.4.19)

.0/

For the coefficients, we have ."/ ! 0 and m1 ! m1 2 .0; 1/ as " ! 0. Using these relations we get from (1.4.19), by evaluating the corresponding limits, that ."/

m1 ."/ =f ."/ ! 1 as " ! 0;

(1.4.20)

which is equivalent to relation (1.4.10). Remark 1.4.2. As in Lemma 1.4.1, the assumption that the limiting distribution function F0 .s/ is non-arithmetic can be replaced in condition D6 by the weaker assumption actually used in the proof that the limiting distribution function F0 .s/ is not concentrated in zero. It follows from (1.4.10) that under the balancing condition B1 , relation (1.4.5) in Theorem 1.4.1 is reduced to the relation (1.3.2) given in Theorem 1.3.1.

1.4.2 Perturbed renewal equations generated by finite measures Consider now, for every "  0, the renewal equation Z t ."/ ."/ x ."/ .t  s/F ."/ .ds/; x .t / D q .t / C 0

t  0;

(1.4.21)

where (a) q ."/ .t / is a function in the class L of (Borel) measurable real-valued functions defined on Œ0; 1/ and bounded on any finite interval, (b) F ."/ .ds/ is a fiC nite measure on B.1/ , the -algebra of Borel subsets of Œ0; 1/, i.e., F ."/ .Œ0; 1// D

F ."/ .1/ < 1, and (c) the measure F ."/ .ds/ is not concentrated in the origin, i.e., F ."/ ..0; 1// > 0. As usual, we denote by F ."/ .t / D F ."/ .Œ0; t /, t  0, the corresponding distribution function generated by the measure F ."/ .ds/, and FN ."/ .t / D F ."/ .t /=F ."/ .1/;

t  0;

f ."/ D 1  F ."/ .1/:

All general facts and definitions concerning renewal equations are translated to the case of the renewal equation (1.4.21) without any changes. P ."/.r / In particular, one can define the renewal function as U ."/ .t / D 1 .t /, rD0 F ."/.r / t  0, where F .t / is the r-fold convolution of the distribution function F ."/ .x/.

62

1

Perturbed renewal equation

The assumption F ."/ ..0; 1// > 0 implies that U ."/ .t /  K1 C K2 t for some finite constants K1 and K2 , i.e., it has not greater than a linear rate of growth in t . Note also that by the definition, U ."/ .t / is a function that is nondecreasing and continuous from the right. The renewal measure U ."/ .A/ on the Borel -algebra on Œ0; 1/ is generated by the renewal function in a usual way. This measure is given by its values on intervals by U ."/ ..a; b/ D U ."/ .b/  U ."/ .a/, 0  a  b < 1; it also has an atom in the point zero. This atom appears due to the 0-fold convolution F ."/.0/ .t /. By the definition, this convolution is a distribution function that has a unit jump in zero. The unique solution of the renewal equation in class L is of the form x ."/ .t / D

Z

t 0

q ."/ .t  s/U ."/ .ds/;

t  0:

We assume the following perturbation condition: D7 :

(a) FN ."/ ./ ) FN .0/ ./ as " ! 0, where FN .0/ .t / is a proper non-arithmetic distribution function; (b) f ."/ ! f .0/ 2 .1; 1/ as " ! 0.

Let us consider the characteristic equation, (1.4.2)  ."/ ./ D 1: If f ."/ < 0 then there exists a unique root of this equation, ."/ < 0. If f ."/ D 0 then there exists a unique root of this equation, ."/ D 0. If f ."/ > 0, then a Cram´er type condition, which requires that 1 <  ."/ .ı/ < 1 for some ı > 0, should be imposed on the distribution functions F ."/ .t / in order to guarantee the existence of a root of the equation (1.4.2). In this case there exists a unique root of this equation, 0 < ."/ < ı. Denote N D f"  0 W f ."/ > 0g: We are interested to ensure the existence of a root of the equation (1.4.2) for all " small enough. Four cases have to be considered: (i) f .0/ > 0; (ii) f .0/ D 0 and 0 is a limit point in the set N, i.e., there exists a sequence 0 < "n ! 0 as n ! 1 such that f ."n / > 0; (iii) f .0/ D 0 and 0 is not a limit point in the set N, i.e., f ."/  0 for all " > 0 small enough; (iv) f .0/ < 0. In the cases (iii) and (iv), no conditions in addition to D7 are required. In the case (ii), condition C1 should be additionally imposed. In the case (i), condition C1 should

1.4 Perturbed renewal equation under Cram´er type conditions

63

be replaced with the following stronger condition: C2 : There exists ı > 0 such that: R1 (a) lim0"!0 0 e ıs F ."/ .ds/ < 1; R1 (b) 0 e ıs F .0/ .ds/ > 1. R1 Note that condition C2 .a/ gives 0 e ıs F .0/ .ds/ < 1, since this corresponds to the case " D 0. If F .0/ .s/ is a proper distribution, then condition C2 .b/ automatically holds. However, if F .0/ .s/ is an improper distribution, then condition C2 .b/ plays an essential role. R1 Let us define ı0 D sup.ı  0 W 0 e ıs F .0/ .ds/ < 1/. If ı0 D 1, then R 1 ıs .0/ e F .ds/ ! 1 as ı ! 1. Thus condition C2 .b/ holds. If ı0 < 1 and R01 ı s .0/ R1 0 F .ds/ D 1, then 0 e ıs F .0/ .ds/ ! 1 as ı < ı0 ; ı ! ı0 . Condition 0 e R1 C2 .b/ again holds. However, if ı0 < 1 and 0 e ı0 s F .0/ .ds/ < 1, then condition C2 .b/ may not hold. For example, let F .0/ .ds/ D Œ1;1/ .t /.1  f /c 1 s 2 e s ds;

R1 where c D 1 s 2 e s ds and f 2 Œ0; 1/. This distribution is improper and R1 .0/ F f . It easy to check that, in this case, ı0 D 1 and 0 e s F .0/ .ds/ D R 1 .1/ D 11 2 s ds D .1  f /c 1 . If .1  f /c 1 < 1, then condition C2 .b/ does 1 .1  f /c not hold. Let us consider a condition weaker than D7 , D8 :

(a) F ."/ ./ ) F .0/ ./ as " ! 0, where F .0/ .t / is a distribution function of a finite measure not concentrated in the origin, i.e., such that F .0/ .0/ < F .0/ .1/; (b) f ."/ ! f .0/ 2 .1; 1/ as " ! 0.

Lemma 1.4.3. Let conditions D8 holds. Then, the following assertions take place: (i) If f .0/ > 0 and condition C2 holds, then there exist ˇ 2 .0; ı and "00 D "00 .ˇ/ > 0 (defined below in formula (1.4.24)) such that  ."/ .ˇ/ 2 .1; 1/ and there exists a unique root 0 < ."/  ˇ of equation (1.4.2) for every "  "00 . (ii) If f .0/ D 0 and 0 is a limit point in the set N, then for any ˇ 2 .0; ı there exists "000 D "000 .ˇ/ > 0 (defined below in formula (1.4.26)) such that  ."/ .ˇ/ 2 .1; 1/ and there exists a unique root ."/ of equation (1.4.2) for every "  "000 , moreover, 0 < ."/  ˇ for " 2 Œ0; "000  \ N, ."/  0 for " 2 Œ0; "000  \ N, and .0/ D 0. (iii) If f .0/ D 0 and 0 is not a limit point in the set N, then there exists "000 0 > 0 (defined below in formula (1.4.27)) such that there exists a unique root ."/  0 .0/ of equation (1.4.2) for every "  "000 D 0. 0 , and 

64

1

Perturbed renewal equation

(iv) If f .0/ < 0, then there exists "0000 0 (defined below in formula (1.4.28)) such that there exists a unique root ."/ < 0 of equation (1.4.2) for every "  "0000 0 . Also, ."/ ! .0/ as " ! 0:

(1.4.22)

Proof. By condition D8 (a), the distribution function F ."/ .s/ is not concentrated in the origin for " small enough, say "  "1 . For "  "1 , the function  ."/ ./ is strongly increasing and continuous on the interval .1; 0. Also,  ."/ .1/ D 0 and  ."/ .0/ D 1  f ."/ . Let us consider the case (i) with f .0/ > 0. In these case, condition C2 .a/ implies that  ."/ .ı/ < 1 for all " small enough, say "  "2 . In this case, the function " ./ is non-negative, strongly increasing, and continuous on the interval .1; ı for every "  min."1 ; "2 /. Also, condition C2 .b/ implies that  .0/ .ı/ > 1. Let us now choose some ˇ 2 .0; ı such that  .0/ .ˇ/ > 1. Now, using condition D8 , we have lim 

"!0

."/

Z .ˇ/  lim lim

T !1 "!1 0

Z

D lim

T !1 0

T

T

e sˇ F" .ds/

e sˇ F0 .ds/ D 0 .ˇ/ > 1:

(1.4.23)

It follows from (1.4.23) that for all " small enough, say "  "3 D "3 .ˇ/, we have " .ˇ/ > 1. Also, " .0/ D 1  f ."/ < 1. The remarks made above imply the existence of a unique positive root of equation (1.4.2) for "  "00 defined by the following formula: "00 D "00 .ˇ/ D min."1 ; "2 ; "3 .ˇ//:

(1.4.24)

Note also that 0 < ."/  ˇ for "  "3 , since " .ˇ/ > 1 and " .0/ D 1  f ."/ < 1 for such ". Let us now consider the case (ii) with f .0/ D 0 and 0 being a limit point in the set N. In this case, the proof is similar to the one given for Lemma 1.4.1. In these case, condition C1 implies that  ."/ .ı/ < 1 for all " small enough, say "  "4 . In this case, the function " ./ is non-negative, strongly increasing, and continuous on the interval .1; ı for every "  min."1 ; "4 /. Because the distribution function F .0/ .s/ is proper and is not concentrated in 0, equation (1.4.2) for the case " D 0 has a unique root .0/ D 0, i.e.,  .0/ .0/ D 1 and  .0/ ./ > 1 for any  2 .0; ı.

1.4 Perturbed renewal equation under Cram´er type conditions

65

Now, using condition D8 we have Z lim " .ı/  lim lim

T

T !1 "!1 0

"!0

Z

D lim

T !1 0

T

e sı F" .ds/

e sı F0 .ds/ D 0 .ı/ > 1:

(1.4.25)

It follows from (1.4.25) that, for any ˇ 2 .0; ı, we have " .ˇ/ > 1 for " small enough, say "  "5 .ˇ/. The remarks made above imply the existence of a unique positive root of equation (1.4.2) for "  "000 defined by the following formula: "000 D "000 .ˇ/ D min."1 ; "4 ; "5 .ˇ//:

(1.4.26)

Note also that 0 < ."/  ˇ for " 2 Œ0; "000 .ˇ/\N, since " .ˇ/ > 1 and " .0/ D 1 f < 1 for such ". Also, ."/  0 for " 2 Œ0; "000 .ˇ/ \ N, since " .0/ D 1  f ."/  1 for such ". Let us consider the case (iii) with f .0/ D 0 and 0 not being a limit point in the set N. In this case,  ."/ .0/ D 1  f ."/  1 for " small enough, say "  "6 . Therefore, there exists a unique non-positive root ."/  0 of equation (1.4.2) for "  "000 0 , where ."/

"000 0 D min."1 ; "6 /:

(1.4.27)

Finally, let us consider the case (iii) with f .0/ < 0. By condition D8 .b/,  ."/ .0/ D 1 f ."/ !  .0/ .0/ D 1 f .0/ > 1 and, therefore, ."/  .0/ D 1  f ."/ > 1 for " small enough, say "  "7 . This implies that there exists a unique negative root ."/ < 0 of equation (1.4.2) for "  "0000 0 , where "0000 0 D min."1 ; "7 /:

(1.4.28)

So, the statement (a) of the lemma is proved. To prove the statement (b) let us again consider four cases. Let f .0/ < 0 or f .0/ D 0 but assume that there is no a limit point in the set N. In these cases, the root ."/  0 for all " small enough. Therefore, D8 gives Z lim lim

1

T !1 "!0 T

e

s."/

F

."/

Z .ds/  lim lim

1

T !1 "!0 T

F ."/ .ds/

D lim lim .F ."/ .1/  F ."/ .T // T !1 "!0

 lim .F .0/ .1/  F .0/ .T =2// D 0: T !1

(1.4.29)

66

1

Perturbed renewal equation

Let f .0/ > 0 or f .0/ D 0 but no limit point exists in the set N. In this case, the fact that ."/ < ˇ < ı for all " small enough yields Z 1 ."/ lim lim e s F ."/ .ds/ T !1 "!0 T

Z

1

 lim lim

T !1 "!0 T

e sˇ F ."/ .ds/

 lim e .ıˇ /T lim

Z

1

"!0 T

T !1

e sı F ."/ .ds/ D 0:

(1.4.30)

Assume that there exists a subsequence "k ! 0 as k ! 1 such that ."k /    , k D 1; 2; : : :, for some  > 0. Then, using D8 , we have Z T ."k / lim lim e s F ."k / .ds/

.0/

T !1 k!1 0

Z

 lim

T

e s.

lim

T !1 k!1 0 Z T s..0/  /

 lim

e

T !1 0

.0/  /

F ."k / .ds/

F .0/ .ds/ D  .0/ ..0/   / < 1:

(1.4.31)

It follows from (1.4.29) or (1.4.30) and (1.4.31) that 1 D lim  ."k / .."k / / k!1

D

lim

Z lim .

T !1 k!1

T 0

e

s."k /

F

."k /

Z .ds/ C

1

T

e s

."k /

F ."k / .ds// < 1:

(1.4.32)

This contradiction shows that no subsequence "k exists, and so lim ."/  .0/ :

"!0

(1.4.33)

Assume now that there exists a subsequence "k ! 0 as k ! 1 such that ."k /  C  , k D 1; 2; : : :, for some  > 0. Then, using D8 we have Z T ."k / 1  lim lim e s F ."k / .ds/

.0/

T !1 k!1 0

Z

 lim

T

lim

T !1 k!1 0

Z

 lim

T !1 0

T

e s.

e s.

.0/ C /

.0/ C /

F ."k / .ds/

F .0/ .ds/ D  .0/ ..0/ C  / > 1:

(1.4.34)

lim ."/  .0/ :

(1.4.35)

This contradiction implies that "!0

The proof of Lemma 1.4.3 is complete.

1.4 Perturbed renewal equation under Cram´er type conditions

67

Let us introduce the condition D9 : F ."/ ./ ) F .0/ ./ as " ! 0, where F .0/ .t / is a distribution function of a finite measure. Lemma formulated below shows that condition D8 (b) in propositions (i) and (ii) of Lemma 1.4.3 can also be omitted. Lemma 1.4.4. Let conditions D9 and C1 hold. Then F ."/ .1/ ! F .0/ .1/ as " ! 0. Proof. Choose 0 < ˇ < ı. Then, using condition C1 , we have Z 1 ."/ ."/ lim lim .F .1/  F .T // D lim lim F ."/ .ds/ T !1 "!0 T !1 "!0 T Z 1  lim lim e ˇs F ."/ .ds/ T !1 "!0 T Z 1 .ıˇ /T  lim e lim e ıs F ."/ .ds/ D 0: T !1

"!0 0

(1.4.36)

Take some sequence 0 < Tk ! 1 as k ! 1 of continuity points of the distribution function F .0/ .s/. Condition D9 implies that F ."/ .Tk / ! F .0/ .Tk / as " ! 0 for every k  1. Obviously, F .0/ .Tk / ! F .0/ .1/ as k ! 1. These relations, together with (1.4.36), imply in an obvious way that lim jF ."/ .1/  F .0/ .1/j

"!0

 lim lim .jF ."/ .1/  F ."/ .Tk /j k!1 "!0

C jF ."/ .Tk /  F .0/ .Tk /j C jF .0/ .Tk /  F .0/ .1/j/ D 0:

(1.4.37)

The proof is complete. Taking into consideration that the limiting characteristic root .0/ may not be equal to zero, we replace condition F5 with the following more general condition: F6 :

(a) limu!0 lim0"!0 supjvju jq ."/ .t C v/  q .0/ .t /j D 0 almost everywhere with respect to the Lebesgue measure on Œ0; 1/; (b) lim0"!0 sup0tT jq ."/ .t /j < 1 for every T  0; P .0/ (c) limT !1 lim0"!0 h r T = h supr ht.rC1/h e . C /t jq ."/ .t /j D 0 for some h > 0 and  > 0.

This condition is used in the cases where (i) f .0/ > 0; or (ii) f .0/ D 0 and 0 is a limit point in the set N; or (iv) f .0/ < 0. Note that, in case (ii), .0/ D 0 and condition F6 coincides with F5 . In the case where (iii) f .0/ D 0 and 0 is not a limit point in the

68

1

Perturbed renewal equation

set N, the weaker condition F2 can be used. Obviously, F6 coincides with condition F2 if to assign  D 0. Denote Z 1 ."/ ."/ m Q1 D se  s F ."/ .ds/: 0

In order to have the relation Q .0/ m Q ."/ 1 !m 1 < 1 as " ! 0;

(1.4.38)

condition C2 or C1 must be added to condition D7 , respectively, in the cases (i) or (ii). In the case (iv), condition D7 automatically implies this relation while, in the case (iii), the following condition has to be added to condition D7 : ."/

.0/

M7 : m1 ! m1 < 1 as " ! 0. Let us define, for "  0, R1 xQ

."/

.1/ D

0

."/

e s q ."/ .s/m.ds/ ."/

m Q1

:

(1.4.39)

The following theorem generalises Theorem 1.4.1 to the case where the total variation of the limiting distribution function F .0/ .1/ can be less, equal, or greater than 1. Theorem 1.4.2. Let condition D7 hold together with the following conditions specified for four alternatives: (i) C2 and F6 if f .0/ > 0; (ii) C1 and F5 if f .0/ D 0 and 0 is a limit point in the set N; (iii) M7 and F2 if f .0/ D 0 and 0 is not a limit point in the set N; (iv) F6 if f .0/ < 0. Then for any 0  t ."/ ! 1 as " ! 0, x ."/ .t ."/ / ! xQ .0/ .1/ as " ! 0: expf."/ t ."/ g

(1.4.40)

Proof. The proof is similar to the proof of Theorem 1.4.1. Let us use the exponential transformation to the renewal equation (1.4.21). Denote Z t ."/ ."/ ."/ ."/ Q e  s F ."/ .ds/; qQ ."/ .t / D e  t q ."/ .t /; xQ ."/ .t / D e  t x ."/ .t /; t  0: F .t / D 0

Note first of all that by the definition of ."/ as a root of equation (1.4.2), it implies that FQ ."/ .1/ D 1, i.e., FQ ."/ .t / is a proper distribution function.

1.4 Perturbed renewal equation under Cram´er type conditions

69

Since x ."/ .t / is a solution of the renewal equation (1.4.21), the function xQ ."/ .t / is a solution of the renewal equation Z t xQ ."/ .t / D qQ ."/ .t / C xQ ."/ .t  s/FQ ."/ .ds/; t  0: (1.4.41) 0

By Lemma 1.4.3,  an obvious way that

."/

!

.0/

as " ! 0. This relation and condition D7 imply in

FQ ."/ ./ ) FQ .0/ ./ as " ! 0;

(1.4.42)

Note also that the limiting distribution function FQ .0/ .s/ is non-arithmetic if the limiting distribution function FN .0/ .s/ is non-arithmetic. Thus the distribution functions FQ ."/ .s/ satisfy condition D2 with the corresponding limiting distribution function FQ .0/ .s/. Also, as was mentioned above, the conditions of the theorem also imply that relation (1.4.38) holds. In the cases (i), (ii), and (iv), since ."/ ! .0/ as " ! 0, we have for any 0 < 1 <  that ."/  .0/ C 1 for all " small enough. Thus the Cram´er type condition C2 implies that Z 1 Z 1 .0/ s."/ ."/ lim lim se F .ds/  lim lim se s. C 1 / F ."/ .ds/ T !1 "!0 T

T !1 "!0 T Z 1 s..0/ C /

 lim

e

T !1 T

F .0/ .ds/ D 0:

(1.4.43)

In the case (iii), since ."/  0 for all " small enough, conditions D7 and M7 imply that Z 1 Z 1 ."/ se s F ."/ .ds/  lim lim sF ."/ .ds/ lim lim T !1 "!0 T

T !1 "!0 T

Z

."/

D lim lim .m1  T !1 "!0

 lim .m.0/ 1 

Z

T !1

T 0

T 0

sF ."/ .ds//

sF .0/ .ds// D 0:

(1.4.44)

Relation (1.4.42), together with (1.4.43) or (1.4.44), imply that Z 1 ."/ ."/ lim m Q 1 D lim se s F ."/ .ds/ "!0

"!0 0

Z

D lim lim

T

T !1 "!0 0 Z T s.0/

D lim

T !1 0

se

."/

se s F ."/ .ds/ .0/

F .0/ .ds/ D m Q1 :

(1.4.45)

70

1

Perturbed renewal equation

Relation (1.4.38) coincides with condition M2 for the distribution functions FQ ."/ .s/. The corresponding limiting moment is m Q .0/ 1 . As was shown above for the cases (i), (ii), and (iv), for any 0 < 1 <  , we have ."/  .0/ C 1 for all " small enough. So ."/ ! .0/ as " ! 0, and condition F6 implies that the functions qQ ."/ .t / satisfy condition F2 . Moreover, for the corresponding R1 .0/ limiting function we have qQ .0/ .t /  e t q .0/ .t / and, therefore, 0 qQ .0/ .s/ ds D R 1 s.0/ .0/ q .s/ ds. 0 e In the case (iii), ."/  0 for all " small enough. Hence, the relation ."/ ! 0 as " ! 0 and condition F2 , assumed to hold for the functions q ."/ .t /, imply that the functions qQ ."/ .t / satisfy condition F2 . Moreover, limiting function R 1 the corresponding R1 satisfies qQ .0/ .t /  q .0/ .t / and, therefore, 0 qQ .0/ .s/ ds D 0 q .0/ .s/ ds (note that .0/ .0/ D 0 in this case and, therefore, e s  1). Now, we can apply Theorem 1.2.1 to equation (1.4.41). Taking into consideration the remarks made above, we get the relation R1 xQ

."/

.t

."/

/De

."/ t ."/ ."/

x

.t

."/

/!

0

.0/

e s q .0/ .s/ ds .0/

m Q1

as " ! 0;

(1.4.46)

which is equivalent to the relation (1.4.40).

1.4.3 Renewal limits for improper perturbed renewal equations Let us see what form will the conditions of Theorem 1.4.2 take if the distribution functions and the forcing function satisfy F ."/ .s/  F .0/ .s/ and q ."/ .s/  q .0/ .s/ for "  0, i.e., they do not depend on the parameter ". The asymptotic relation (1.4.40) takes in this case the following form: x .0/ .t / ! xQ .0/ .1/ as " ! 0: expf.0/ t g

(1.4.47)

Condition D7 reduces to the assumption that FN .0/ .t / is a non-arithmetic distribution function. R1 In case (i), condition C2 reduces to the assumption that 0 e ıs F .0/ .ds/ 2 .1; 1/ .0/ for some ı > 0, and condition F6 to the assumption that the function e . C /s q .0/ .s/ is directly Riemann integrable on Œ0; 1/ for some  > 0. Note that .0/ > 0 in this case. These conditions (together with the reduced version of condition D7 ) coincide with the conditions of the refined generalisation of the renewal Theorem 1.1.1 to the model of an improper renewal equation given in Feller (1966). The case (ii) is impossible, which is easily seen, since N D ¿. R1 .0/ In the case (iii), condition M7 reduces to the assumption m1 D 0 sF .0/ .ds/ < 1, and condition F2 to the assumption that the function q .0/ .s/ is directly Riemann

1.4 Perturbed renewal equation under Cram´er type conditions

71

integrable on Œ0; 1/. Together with the reduced version of condition D7 , these conditions are the standard in the renewal Theorem 1.1.1. Note that .0/ D 0 in this case. In the case (iv), condition F6 reduces to the assumption that the function .0/ e . C /s q .0/ .s/ is directly Riemann integrable on Œ0; 1/ for some  > 0. Note that, in this case, .0/ < 0. If the conditions listed above are imposed on characteristics of the perturbed renewal equation by formally replacing F .0/ .s/ and q .0/ .s/ with F ."/ .s/ and q ."/ .s/, then the asymptotic relation (1.4.47) would take place if 0 is formally replaced with ". The following question arises. Do the conditions of Theorem 1.4.2 guarantee that they remain true for all " small enough? The answer is affirmative for moment type conditions. Condition C2 implies, under R1 condition D7 , that 0 e ıs F ."/ .ds/ 2 .1; 1/ for all " small enough, say "  "1 . ."/ Also, condition M7 implies, under condition D7 , that m Q 1 < 1 for all " small enough, say "  "2 , in the case (iii). The answer is negative for conditions from classes D and F. Non-arithmetic property of the limiting distribution function FN .0/ .s/ does not guarantee that the prelimiting distribution functions FN ."/ .s/ also possess this property. Also, the direct Rie.0/ mann integrability of the limiting forcing function e . C /s q .0/ .s/ does not guarantee the same property for the corresponding pre-limiting functions. The following two conditions should be assumed: D10 : FN ."/ .s/ is a non-arithmetic distribution function for all " small enough, say "  "3 ; and ."/

F7 : The function e . C /s q ."/ .s/ is directly Riemann integrable on Œ0; 1/ for all " small enough, say "  "4 . Note that, under conditions of Theorem 1.4.2, it is enough to assume that q ."/ .s/ is a continuous function almost everywhere with respect to the Lebesgue measure on Œ0; 1/. In this case, condition F7 holds and the Lebesgue integration in the right-hand side of (1.4.39) can be replaced with the Riemann integration. Define "0 D min."1 ; "2 ; "3 ; "4 /:

(1.4.48)

The following theorem follows from the above remarks. Theorem 1.4.3. Let conditions of Theorem 1.4.2 as well as conditions D10 and F7 hold. Then, for all "  "0 , xQ ."/ .t / D e 

."/ t

x ."/ .t / ! xQ ."/ .1/ as t ! 1:

(1.4.49)

72

1

Perturbed renewal equation

Let us remark that, under conditions of Theorem 1.4.2, the expression for xQ ."/ .1/ given by formula (1.4.39) is finite for all "  0 small enough without the assumption that q ."/ .s/ is a function continuous almost everywhere with respect to the Lebesgue measure on Œ0; 1/. This is so, since the Lebesgue integration is used in this formula. Moreover, it can be proved that the mean value of xQ ."/ ./ over the interval Œ0; t  converges to xQ ."/ .1/ as t ! 1.

1.4.4 Convergence of renewal limits for improper perturbed renewal equations The following theorem supplements the asymptotic relations given in relations (1.4.40) and (1.4.49). Theorem 1.4.4. Let conditions of Theorem 1.4.2 hold. Then xQ ."/ .1/ ! xQ .0/ .1/ as " ! 0:

(1.4.50)

Proof. As was shown in the proof of Theorem 1.4.2, the conditions of Theorem 1.4.2 imply that ."/ ! .0/ as " ! 0; condition M2 holds for the distributions FQ ."/ .t / D R t s."/ ."/ ."/ F .ds/; and condition F2 holds for the functions qQ ."/ .s/ D e s q ."/ .s/. 0 e Hence, relation (1.4.50) follows from Theorem 1.2.4.

Chapter 2

Nonlinearly perturbed renewal equation

Chapter 2 contains results concerning exponential expansions for solutions of the perturbed renewal equation for the model in which some Cram´er type condition is imposed on the distribution that generates the perturbed renewal equation. Some additional nonlinear perturbation conditions are also imposed on the distributions. These conditions mean that the moments of these distributions are nonlinear functions of the perturbation parameter " and can be expanded in an asymptotic power series with respect to " up to and including some order k. Theorems in Chapter 1 give exponential normalising functions in an implicit form as a solution of the nonlinear characteristic equation (1.0.7). Further improvements should seal with a more explicit form for the exponential normalising functions in the asymptotic relation (1.0.5). Theorems in Chapter 2 give expressions for the corresponding coefficients in exponential expansions of the corresponding exponential normalising functions in explicit form that makes this asymptotical relation much more useful and effective. A new improvement is achieved under natural additional perturbation conditions in which we assume that the defect f ."/ and the corresponding moments of the distribution F ."/ .s/ can be expanded in power series with respect to " up to and including an order k. Under these perturbation conditions, we give an asymptotic expansion for the characteristic roots ."/ , ."/ D .0/ C a1 " C    C ak "k C o."k /;

(2.0.1)

and an explicit algorithm for calculating the coefficients .0/ , a1 ; a2 ; : : :, in the expansions of the corresponding perturbation conditions. Asymptotic expansion (2.0.1) permits to introduce various balancing conditions between the rate of growth of time t ."/ ! 1 and the perturbation rate determined by the perturbation parameter " ! 0. These conditions are controlled by an integer parameter r, 1  r  k, and have the following form: "r t ."/ ! r as " ! 0:

(2.0.2)

For some of these balancing conditions holds, the asymptotics (1.0.5) can be represented in the following form: x ."/ .t ."/ / ! e r ar xQ .0/ .1/ as " ! 0: (2.0.3) expf..0/ C a1 " C    C ar 1 "r 1 /t ."/ g

74

2 Nonlinearly perturbed renewal equation

Each quantity expf.0/ t ."/ g, expfa1 "t ."/ g; : : : ; expfar1 "t ."/ g contributes to the normalising function in the left-hand side of (2.0.2) a factor that tends to zero, equal to 1, or tends to 1 depending on whether the corresponding coefficients .0/ ; a1 ; : : : ; ar 1 take positive, zero, or negative values. Also, as was mentioned above, for the asymptotic relation (2.0.2) there is an explicit algorithm for calculating the coefficients .0/ , a1 ; : : : ; ar 1 . This makes the asymptotics given in (2.0.2) much more explicit and effective than (1.0.5). Theorems 2.1.1, 2.1.2, and 2.1.3 yield explicit exponential expansions for solutions of asymptotically proper perturbed renewal equations while Theorems 2.2.1 and 2.2.2 yield similar results for asymptotically improper perturbed renewal equations. They extend the results in Theorems 1.4.1 and 1.4.2. Finally in Theorem 2.3.2 we give asymptotical expansions for renewal limits of nonlinearly perturbed renewal equations. The asymptotics of type (2.0.2) is a basis for studies of the so-called mixed ergodic and limit theorems and mixed ergodic and large deviation theorems for perturbed stochastic processes with absorption. These studies are carried out in Chapters 3–5.

2.1 Exponential expansions for perturbed renewal equation In this section, we give exponential asymptotic expansions for solutions of perturbed asymptotically proper renewal equations.

2.1.1 Exponential expansions based on defects of distributions that generate perturbed renewal equations Consider, for every "  0, the renewal equation Z t ."/ ."/ x ."/ .t  s/F ."/ .ds/; x .t / D q .t / C 0

t  0;

(2.1.1)

where (a) q ."/ .t / is a measurable and locally bounded real-valued function on Œ0; 1/, and (b) F ."/ .s/ is a distribution function on Œ0; 1/ which is not concentrated at 0 but can be improper, i.e., F ."/ .1/ 2 .0; 1/. As was mentioned in Chapter 1, there is a unique measurable and locally bounded solution x ."/ .t / of equation (2.1.1). We use the notations Z 1 ."/ ."/ ."/ f D 1  F .1/; mr D s r F ."/ .ds/; r  1: 0

Let us recall the condition (introduced in Subsection 1.3.1): D6 : F ."/ ./ ) F .0/ ./ as " ! 0, where F .0/ .s/ is a proper non-arithmetic distribution function.

2.1 Exponential expansions for perturbed renewal equation

75

Condition D6 includes the assumption that F .0/ .1/ D 1, which automatically implies the following relation: f ."/ ! f .0/ D 0 as " ! 0:

(2.1.2)

Now, we assume the following Cram´er type condition (introduced in Subsection 1.4.1): R1 C1 : There exists ı > 0 such that lim0"!0 0 e ıs F ."/ .ds/ < 1. Let us note that conditions D6 and C1 imply that m."/ ! m.0/ r r 2 .0; 1/ as " ! 0; Z

Denote 

."/

./ D

1 0

r  1:

(2.1.3)

e s F ."/ .ds/:

It follows from condition C1 that there exists "1 > 0 such that  ."/ .ı/ < 1 for ."/ "  "1 . Obviously, mr < 1, r D 1; 2; : : :, for "  "1 , as well. The following plays a crucial role in the further consideration:  ."/ ./ D 1:

(2.1.4)

It follows from Lemma 1.4.1 that, under conditions D6 and C1 , there exists "2 > 0 such that equation (2.1.4) has a unique nonnegative root ."/ for "  min."1 ; "2 /. Also, ."/ ! .0/ D 0 as " ! 0:

(2.1.5)

We also assume the following condition (introduced in Subsection 1.4.1): F5 :

(a) limu!0 lim0"!0 supjvju jq ."/ .t C v/  q .0/ .t /j D 0 almost everywhere with respect to Lebesgue measure on Œ0; 1/; (b) lim0"!0 sup0tT jq ."/ .t /j < 1 for every T  0; P (c) limT !1 lim0"!0 h r T = h supr ht.rC1/h e t jq ."/ .t /j D 0 for some h > 0 and some  > 0.

Let us also denote

R1 x

."/

.1/ D

0

q ."/ .s/m.ds/ ."/

:

m1

We shall also use the following balancing condition: .k/

B2 : 0  t ."/ ! 1 and f ."/ ! 0 as " ! 0 in such a way that .f ."/ /k t ."/ ! k , where 0  k < 1.

76

2 Nonlinearly perturbed renewal equation

The following theorem generalises Theorem 1.4.1. Theorem 2.1.1. Let conditions D6 and C1 hold. Then (i) The following asymptotic expansion holds for the root ."/ of the equation (2.1.4) for every k  1: ."/ D b"1 f ."/ C    C b"k .f ."/ /k C o..f ."/ /k /;

(2.1.6)

where the coefficients b"k , k  1, can be calculated using the recurrence for."/ 1 ."/ 2 1 1 mulas b"1 D .m."/ 1 / , b"2 D  2 .m1 / m2 b"1 and, in general, for k  2, ."/

b"k D  .m1 /1

k X

X

m."/ r 

r D2

k1 Y

ni b"i =ni Š;

(2.1.7)

n1 ;:::;nk1 2Dr;k iD1

where Dr;k , for every 2  r  k < 1, is the set of all nonnegative, integer solutions of the system n1 C    C nk1 D r;

n1 C    C .k  1/nk1 D k:

(2.1.8)

.k/

(ii) If, additionally, conditions B2 , for some k  1, and F5 hold then the following asymptotic relation takes place:

expf.b"1 f

."/

x" .t ."/ / C    C b"k1 .f ."/ /k1 /t ."/ g

! e k b0k x0 .1/ as " ! 0:

(2.1.9)

Proof. As was mentioned above, condition C1 implies that  ."/ .ı/ < 1 for "  "1 , and Lemma 1.4.1 implies that, under conditions D6 and C1 , there exists a unique nonnegative root ."/ of equation (2.1.4) for "  min."1 ; "2 / and that ."/ ! 0 as " ! 0. Apply Taylor’s expansion to the function e t , which can be given in the following form (using the fact that this function and its derivatives are increasing functions): e s

."/

D 1 C s."/ =1Š C    C s k .."/ /k =kŠ ."/

."/ C s kC1 .."/ /kC1 e s kC1 .s/=.k C 1/Š;

(2.1.10)

."/ .s/; s  0, is a continuous function such that where kC1 ."/

0  kC1 .s/  1;

s  0:

(2.1.11)

2.1 Exponential expansions for perturbed renewal equation

77

As was shown in the proof of Lemma 1.4.1, for any ˇ 2 .0; ı there exists "3 D "3 .ˇ/ > 0 such that "  ˇ for "  "3 .ˇ/. By condition C1 , Z 1 Z 1 kC1 ˇs ."/ lim s e F .ds/  lim ckC1 e ıs F ."/ .ds/ < 1; (2.1.12) 0"!0 0

0"!0

0

where ckC1 D sups0 s kC1 e .ıˇ /s < 1. It follows from relation .2:1:12/ that there exists "4 D "4 .ˇ/ > 0 such that, for "  "4 , Z 1 1 s kC1 e sˇ F ."/ .ds/ < 1: (2.1.13) MkC1 D sup .k C 1/Š ""4 0 Let us define "5 D "5 .ˇ/ D min."1 ; "2 ; "3 .ˇ/; "4 .ˇ//:

(2.1.14)

Integrating both sides in (2.1.10) and using (2.1.11) and (2.1.13) we get, for "  "5 , Z 1 ."/ ."/ e s F ."/ .ds/ D 1  f ."/ C m."/ 1  =1Š C    0

."/

."/

C mk .."/ /k =kŠ C .."/ /kC1 MkC1 kC1 ;

(2.1.15)

where R1

."/ kC1

."/

."/

s kC1 e s kC1 .s/F ."/ .ds/ R1 D 2 Œ0; 1: sup""4 0 s kC1 e sˇ F ."/ .ds/ 0

(2.1.16)

Therefore, equation (2.1.4) can be rewritten in the form ."/

."/

."/

m1 ."/ =1Š C m2 .."/ /2 =2Š C    C mk .."/ /k =kŠ ."/

C .."/ /kC1 MkC1 kC1 D f ."/ :

(2.1.17)

The coefficients ."/ ! 0, and the sum of all terms in the left-hand side in (2.1.17), .0/ beginning with the second one, is o.."/ /. Also, m."/ r ! mr 2 .0; 1/. Using these facts, we get from (2.1.17), dividing both sides by f ."/ and evaluating the correspond."/ ing limits, that m1 ."/ =f ."/ ! 1 as " ! 0. This means, in equivalent terms, that the coefficients ."/ can be represented in the form ."/

."/ D b"1 f ."/ C 1 ; ."/

."/

where b"1 D .m1 /1 and 1 D o.f ."/ /. Relation (2.1.18) becomes (2.1.6) for the case k D 1.

(2.1.18)

78

2 Nonlinearly perturbed renewal equation ."/

Substituting (2.1.18) into (2.1.17) and cancelling the terms m1 ."/ D f ."/ on the left and the right-hand sides we can rewrite equation (2.1.17) in the form suitable for asymptotic analysis of the residual function "1 , ."/ ."/

."/

."/

m1 1 =1Š C m2 .b"1 f ."/ C 1 /2 =2Š C    ."/

."/

."/

."/

C mk .b"1 f ."/ C 1 /k =kŠ C .b"1 f ."/ C 1 /kC1 MkC1 kC1 D 0: (2.1.19) Using ."/ ! 0 and since 1."/ D o.f ."/ / we get from (2.1.19), dividing both ."/ ."/ sides by .f ."/ /2 and evaluating the corresponding limits, that m1 1 =.f ."/ /2 C ."/ ."/ 2 b"1 m2 =2 ! 0 as " ! 0. This equivalently means that the coefficients 1 can be represented in the form ."/

1 D b"2 .f ."/ /2 C "2 ;

(2.1.20)

1 1 m."/ b 2 and ."/ D o..f ."/ /2 /. where b"2 D .m."/ 1 / 2 2 2 "1 Relation (2.1.20) is equivalent to relation (2.1.6) for the case k D 2. Continuing in the same way we can obtain relation (2.1.6) for the cases k > 2 as well. Note also that the values of the coefficients b"r , r  1, do not depend on the order k chosen in expansion (2.1.6). Because it is now known that there exists asymptotical expansion (2.1.6), the algorithm described above allows to get the coefficients b"k by collecting and setting to 0 the coefficients for f k , k  1, in the formal asymptotic equality that follows from equation (2.1.17), ."/

m1 .b"1 f C b"2 f 2 C    /=1Š ."/

C m2 .b"1 f C b"2 f 2 C    /2 =2Š C    D f:

(2.1.21)

."/

In the case k D 1, this equality yields the equation m1 b"1 D 1 and, if k  2, the equation ."/

m1 b"k C

k X r D2

m."/ r 

X

k1 Y

ni b"i =ni Š D 0;

(2.1.22)

n1 ;:::;nk1 2Dr;k iD1

which leads to the general recurrence formula (2.1.7) for calculating the coefficients b"k . The statement (i) of Theorem 2.1.1 is proved. Let us now use Theorem 1.4.1. This theorem implies that, under conditions D6 , C1 , and F5 , for any 0  t ."/ ! 1 the following asymptotic relation holds: x" .t ."/ / ! x0 .1/ as " ! 0: expf."/ t ."/ g

(2.1.23)

2.1 Exponential expansions for perturbed renewal equation

79

The statement (ii) is a corollary of relations (2.1.6) and (2.1.23). Under the condi."/ k tion .f ."/ /k t" ! k 2 Œ0; 1/, we have o..f ."/ /k /t" ! 0, and so e o..f / /t" ! 1. Also, b"k .f ."/ /k t" ! b0k k . Using these facts one can easily transform relation (2.1.23) into the form (2.1.9).

2.1.2 Asymptotically proper perturbed renewal equations with the defect of order O."/ ."/

Due to convergence of the moments mr , we have b"r ! b0r as " ! 0;

r  1:

Nevertheless, it is not convenient to have the coefficients in the exponential asymptotic relation (2.1.9) dependent of perturbation parameter ". This situation can be ."/ improved if the probabilities f ."/ and the moments mr can be expanded in asymptotic power series with respect to the perturbation parameter ". In this case, expansion (2.1.6) and relation (2.1.9) can be rewritten in the form such that the corresponding coefficients would not depend on ". Let us assume that the following perturbation condition holds for some k  1: P1.k/ :

(a) 1  f ."/ D 1 C b1;0 " C    C bk;0 "k C o."k /, where jbr;0 j < 1, r D 1; : : : ; k; ."/

.0/

(b) mr D mr C b1;r " C    C bkr;r "kr C o."kr / for r D 1; : : : ; k, where jbl;r j < 1, l D 1; : : :, k  r, r D 1; : : : ; k. .0/

It is convenient to also set the coefficients b0;0 D 1 and b0;r D mr , r  1. R1 ."/ Let us also remark that the identity 1  f ."/ D F ."/ .1/ D 0 F ."/ .dx/ D m0 can be interpreted as a moment functional of the same type as the moment functionals R 1 r ."/ m."/ .dx/, r  1, but of order r D 0. This remark makes the expansions r D 0 x F given in conditions P1.k/ (a) and (b) consistent. .k/

Condition P1 should be interpreted as a nonlinear perturbation condition imposed on the moment functionals of the distribution functions that generate the corresponding renewal equations. These conditions mean that the moment functionals m."/ r , r D 0; : : : ; k, of these distribution functions are nonlinear smooth functions of the ."/ perturbation parameter " such that mr can be expanded in an asymptotic power series with respect to " up to and including the order k  r for every r D 0; : : : ; k. It follows from condition D6 that f ."/ ! 0 as " ! 0, which is consistent with .k/ .k/ condition P1 .a/. The first non-zero coefficient br;0 in P1 .a/ must be negative, since, by the definition, 1  f ."/  1. Note also that this condition allows for the case where br;0 D 0 for all r D 1; : : : ; k.

80

2 Nonlinearly perturbed renewal equation ."/

It follows from conditions D6 and C1 that the power moments mr are finite for all r  1 and all " small enough, and that .0/ m."/ r ! mr 2 .0; 1/ as " ! 0;

r  1;

(2.1.24)

which agrees with condition P1.k/ .b/. We shall also use the following balancing condition for 1  r  k: .r/

B3 : 0  t ."/ ! 1 as " ! 0 in such a way that "r t ."/ ! r , where 0  r < 1. .k/

Theorem 2.1.2. Let conditions D6 , C1 and P1 hold. Then (i) For all " small enough, equation (2.1.4) has a unique nonnegative root ."/ that has the following asymptotic expansion: ."/ D a1 " C    C ak "k C o."k /;

(2.1.25)

where the coefficients an , n D 1; : : : ; k, are given by the recurrence formulas a1 D b1;0 =b0;1 and, in general, for n D 1; : : : ; k,  n1 X 1 bn;0 C bnq;1 aq an D b0;1 qD1

C

n X n X mD2 qDm

bnq;m 

X

q1 Y



n app =np Š

; (2.1.26)

n1 ;:::;nq1 2Dm;q pD1

where Dm;q , for every 2  m  q < 1, is the set of all nonnegative, integer solutions of the system n1 C    C nq1 D m; n1 C    C .q  1/nq1 D q:

(2.1.27)

(ii) If the coefficients satisfy bl;0 D 0, l D 1; : : : ; r, for some 1  r  k, then a1 ; : : : ; ar D 0. If bl;0 D 0, l D 1; : : : ; r  1, but br;0 < 0 for some 1  r  k, then a1 ; : : : ; ar 1 D 0 but ar > 0. (iii) If, additionally, conditions B.r/ 3 , for some 1  r  k, and F5 hold, then the following asymptotic relation holds: x ."/ .t ."/ / ! e r ar x .0/ .1/ as " ! 0: (2.1.28) expf.a1 " C    C ar 1 "r 1 /t ."/ g Proof. The proof follows the scheme used in the proof of Theorem 2.1.1. It was proved in Lemma 1.4.1 that under conditions D6 and C1 there exists "5 > 0, defined in formula (2.1.14), such that equation (2.1.4) has a unique nonnegative root ."/ for all "  "5 , and that ."/ ! 0 as " ! 0.

2.1 Exponential expansions for perturbed renewal equation

81

Now, we start with an asymptotic analysis for the coefficients ."/ . We use equation (2.1.17) obtained, for "  "5 , in the proof of Theorem 2.1.1, ."/ ."/ 2 ."/ m."/ 1  =1Š C m2 . / =2Š C    ."/ C m."/ .."/ /k =kŠ C .."/ /kC1 MkC1 kC1 D f ."/ ; k

(2.1.29)

."/ 2 Œ0; 1. where MkC1 < 1 and kC1 ."/ The coefficients  ! 0, and the sum of all terms in the left-hand side of (2.1.29), ."/ .0/ starting with the second one, is o.."/ /. Also, mr ! mr 2 .0; 1/ for r  1. Note also that the first term in the asymptotic expansion for the defect f ."/ , as implied .k/ by condition P1 (a), is of the order O."/, more precisely, it equals b1;0 ". Thus, dividing both sides of (2.1.29) by m."/ 1 " and evaluating the corresponding limits we .k/ ."/ obtain using P1 that  =" ! b1;0 =b0;1 . This means that ."/ can be represented in the form ."/

."/ D a1 " C 1 ;

(2.1.30)

."/

where a1 D b1;0 =b0;1 and 1 D o."/. Relation (2.1.30) reduces to (2.1.25) in the case k D 1. .k/ Let k  2. Substituting the expansions given in condition P1 and (2.1.30) into (2.1.29) and taking into account the identity a1 D b1;0 =b0;1 , we get .b1;1 " C o."//a1 " C .b0;1 C b1;1 " C o."//1."/ C .b0;2 C o.1//.a1 " C 1."/ /2 =2Š C    C .b0;k C o.1//.a1 " C 1."/ /k =kŠ D b2;0 "2 C o."2 /:

(2.1.31)

Dividing both sides in (2.1.31) by b0;1 "2 and evaluating the corresponding limits we ."/ obtain that b1;1 a1 =b0;1 C 1 ="2 C b0;2 a12 =2b0;1 ! b2;0 =b0;1 . This is equivalent to the following asymptotical relation: ."/

."/

1 D a2 "2 C 2 ;

(2.1.32)

."/ 1 1 .b 2 2 where a2 D b0;1 2;0 C b1;1 a1 C 2 b0;2 a1 / and 2 D o." /. Relations (2.1.30) and (2.1.32) yield relation (2.1.25) for k D 2. The expression for a2 given above is exactly formula (2.1.26) for k D 2. Repeating the above argument in the general case we obtain expansion (2.1.25) and formula (2.1.26) for k > 2. However, formula (2.1.26) can be obtained in a simpler way once the asymptotic expansion (2.1.25) has already been obtained. From (2.1.29) we get the following formal equation:

.b0;1 C b1;1 " C    /.a1 " C a2 "2 C    /=1Š C .b0;2 C b1;2 " C    /.a1 " C a2 "2 C    /2 =2Š C    D  .b1;0 " C b2;0 "2 C    /:

(2.1.33)

82

2 Nonlinearly perturbed renewal equation

Equating the coefficients of "n for n  1 in the right and the left-hand sides of (2.1.33) we obtain formula (2.1.26) for calculating the coefficients a1 ; : : : ; ak . The first summand in (2.1.26) reflects the contribution of the sum in the right-hand side of (2.1.33). The second and the third terms in the sum in (2.1.26) equal 0 for n D 1 and, in this case, formula (2.1.33) yields values of the coefficients a1 D b1;0 =b0;1 . Thus the only case where n  2 should be considered. The second term in (2.1.26) reflects the contribution of the first product of sums to the left-hand side in (2.1.33). The sum with the index 2  m  n in (2.1.26) shows the contribution of the product of the sums .b0;m C    /.a1 " C    /m =mŠ in the right-hand side in (2.1.33). In this case, we collect all terms in this product such that the sum .b0;m C    / contributes "nq and .a1 " C    /m =mŠ contributes "q for m  q  n. Note that the cases 1  q < m are impossible, since the minimal degree of the power of " in .a1 " C    /m =mŠ is m. The coefficient at "q , coming from .a1 " C    /m =mŠ, is the sum of n the products a1n1    aq q , where the nonnegative integer coefficients n1 ; : : : ; nq must satisfy the obvious relations (a) n1 C    C nq D m and (b) n1 C    C q nq D q. These relations reduce to (2.1.27), since the coefficient nq can take the only value 0. Indeed, relation (b) implies that the only values 0 and 1 are possible for the coefficient nq . However, if nq D 1, then all other coefficients n1 ; : : : ; nq1 must equal 0. In this case, n1 C    C nq D 1, which contradicts the restriction 2  m  n. Therefore, the coefficient nq must be 0. n Finally, the combinatoric counting of the coefficient at the product a1n1    aq q yields the number mŠ=n1 Š    nq Š. The coefficient mŠ cancels since it also appears as the denominator of .a1 " C    /m =mŠ. Also, nq D 0, as was mentioned above. This completes the proof of (i). The statement (ii) is a direct corollary of the recursive formula (2.1.26). Let us use now Theorem 1.4.1. This theorem implies that under conditions D6 , C1 and F5 for any 0  t ."/ ! 1 the following asymptotical relation holds: x ."/ .t ."/ / ! x .0/ .1/ as " ! 0: ."/ ."/ expf t g

(2.1.34)

Statement (iii) is an obvious corollary of relations (2.1.25) and (2.1.34). Under condition "r t" ! r 2 Œ0; 1/, we have .ar C1 "rC1 C    C ak "k C o."k //t" ! 0 and, so, expf.ar C1 "r C1 C    C ak "k C o."k //t" g ! 1. Also, ar "r t" ! ar r . Using these facts and the asymptotic expansion ."/ D a1 " C    C ak "k C o."k / obtained in (i), one can easily transform relation (2.1.34) to the form (2.1.28). The proof is complete. Remark 2.1.1. The coefficient an can be calculated by using formula (2.1.26) if n  k and the value of coefficient an does not depend on the parameter k. Formula (2.1.26) gives the following explicit expressions for first three coefficients in the

83

2.1 Exponential expansions for perturbed renewal equation

expansion (2.1.25) in terms of the coefficients that appear in the perturbation condition P1.k/ : a1 D 

b1;0 ; b0;1

1 b2;0 b1;1 b1;0 .b2;0 C b1;1 a1 C b0;2 a12 / D  C  2 b0;1 2 b0;1 b0;1 1  1 b3;0 C b2;1 a1 C b1;1 a2 C b1;2 a12 C b0;2 a1 a2 C a3 D  b0;1 2 a2 D 

1

2 b0;2 b1;0 3 2b0;1

;

 1 b0;3 a13 : (2.1.35) 6

2.1.3 Asymptotically proper perturbed renewal equation with the defect of order O."h / Let us introduce integer parameters 1  h  k0 and kr  1, r D 1; : : :, and define a vector parameter kN D .h; k0 ; : : : ; kNQ /, where NQ D min.n W nh  k0 /. We assume that the following perturbation condition holds: .kN /

P2 :

(a) 1  f ."/ D 1 C bh;0 "h C    C bk0 ;0 "k0 C o."k0 /, where jbl;0 j < 1, l D h; : : : ; k0 ; ."/ .0/ (b) mr D mr C b1;r " C    C bkr ;r "kr C o."kr / for r D 1; : : : ; NQ , where jbl;r j < 1; l D 1; : : : ; kr , r D 1; : : : ; NQ . .0/

As before, it is convenient to also define the coefficients b0;0 D 1 and b0;r D mr , r  1. .kN / .k/ Condition P2 is more general than condition P1 , and the first one reduces to the second one if h D 1; k0 D k (in this case, NQ D k) and kr D k  r, r D 1; : : : ; NQ . The following theorem supplements and generalises Theorem 2.1.2. .kN /

Theorem 2.1.3. Let conditions D6 , C1 , and P2 hold. Then (i) For all " small enough, equation (2.1.4) has a unique nonnegative root ."/ with the following asymptotic expansion: ."/ D ah "h C    C ak "k C o."k /;

(2.1.36)

where k D min.k0 ; k1 Ch; : : : ; kNQ ChNQ / and the coefficients an , n D h; : : : ; k, are given by the recurrence formulas ah D bh;0 =b0;1 and, in general, for n D h; : : : ; k,  n1 X 1 an D b0;1 bn;0 C bni;1 ai iDh

C

Œn= Xh

n X

r D2 j Dr h

bnj;r 

X

jY 1

nh ;:::;nj 1 2Dh;r;j iDh

 ; (2.1.37)

aini =ni Š

84

2 Nonlinearly perturbed renewal equation

where Dh;r;j , for every h  1 and 2  r  j < 1, is the set of all nonnegative, integer solutions of the system nh C    C nj 1 D r; hnh C    C .j  1/nj 1 D j:

(2.1.38)

(ii) If bl;0 D 0, l D h; : : : ; r, for for some h  r  k, then a1 ; : : : ; ar D 0. If bl;0 D 0, l D h; : : : ; r  1 but br;0 < 0 for some h  r  k, then ah ; : : : ; ar 1 D 0 but ar > 0. .r/

(iii) If, additionally, conditions B3 , for some h  r  k, and F5 hold, then the following asymptotic relation holds: x ."/ .t ."/ / ! e r ar x .0/ .1/ as " ! 0: (2.1.39) expf.ah " C    C ar 1 "r 1 /t ."/ g Proof. The first part of the proof that yields equation (2.1.29) is the same as in the proof of Theorem 2.1.2. We have ."/ ! 0, and the sum of all terms in the left-hand side of (2.1.29), starting .0/ with the second one, is o.."/ /. Also, m."/ 2 .0; 1/ for r  1. However, r ! mr .kN / the first term in the asymptotic expansion for the defect f ."/ given by condition P2 (a) is not of order O."/, but of the order O."h /, more precisely it equals bh;0 "h . ."/ Dividing both sides in (2.1.29) by m1 "h and evaluating the corresponding limits we .kN /

obtain using P2 that ."/ ="h ! bh;0 =b0;1 . This means that ."/ can be represented in the form ."/ D ah " C h."/ ;

(2.1.40)

where ah D bh;0 =b0;1 and h."/ D o."h /. Relation (2.1.40) reduces to (2.1.36) if k D h. The following steps in the proof of existence of the corresponding expansion for the root ."/ are similar to the ones in the proof of Theorem 2.1.2. Let k  hC1. This means that k0 ; k1 Ch; : : : ; kNQ CNQ h  hC1. Thus, substituting .kN /

the expansions given in condition P2 and (2.1.40) into (2.1.29) and using the identity ah D bh;0 =b0;1 one can rewrite (2.1.29) in the following form: ."/

.b1;1 " C o."//ah "h C .b0;1 C b1;1 " C o."//h

C .b0;2 C o.1//.ah "h C h."/ /2 =2Š C    ."/

C .b0;k C o.1//.ah "h C h /k =kŠ D  bhC1;0 "hC1 C o."hC1 /:

(2.1.41)

Then, divide both sides in (2.1.41) by b0;1 "hC1 and evaluate the corresponding limits. The terms b0;r ahr "r h =b0;1 "hC1 for r  2 can contribute with non-zero limits

2.1 Exponential expansions for perturbed renewal equation

85

only if 2  r  Œ.h C 1/= h. In fact, this is possible only if h D 1. We thus obtain, by evaluating the corresponding limits, that b1;1 ah =b0;1 C 1."/ ="hC1 C .2  r  Œ.h C 1/= h/b0;2 ah2 =2b0;1 ! bhC1;0 =b0;1 . This is equivalent to the following asymptotical relation: ."/

."/

1 D ahC1 "hC1 C nC1 ;

(2.1.42)

1 where ahC1 D b0;1 .bhC1;0  b1;1 ah  .2  r  Œ.h C 1/= h/b0;2 ah2 =2 and

."/ D o."hC1 /. hC1 Relations (2.1.40) and (2.1.42) yield the relation (2.1.36) for the case k D h C 1. The expression for a2 given above is exactly formula (2.1.37) with k D h C 1. Repeating the above argument in the general case we obtain expansion (2.1.36) and formula (2.1.37) for k > h C 1. Let us make a comment on the interruption procedure for this algorithm. The algorithm should stop at the point where the limits in (2.1.41) can be evaluated, after the corresponding exclusion transformation and division by b0;1 "n were performed, but can not be evaluated for b0;1 "nC1 . This will happen when we find for first time (by increasing n) that "nC1 can not be compared with at least one of the remaining terms Q of the form o."k0 /; o."k1 /"h ; : : : ; o."kN /"N h , where o."k0 /; o."k1 /; : : : ; o."kNQ / are N the corresponding remaining terms in the perturbation expansions in condition P.2k/ . This will occur for n D k. Formula (2.1.37) can be obtained from the asymptotic expansion (2.1.36) in a way similar to the one used in the proof of Theorem 2.1.2. The coefficients ah ; : : : ; ak can be found by equating the coefficients of "n for n  1 on the right and the left-hand hand sides of the following formal relation obtained from (2.1.29):

.b0;1 C b1;1 " C    /.ah "h C ahC1 "hC1    /=1Š C .b0;2 C b1;2 " C    /.ah "h C ahC1 "hC1 C    /2 =2Š C    D .bh;0 "h C bhC1;0 "hC1 C    /:

(2.1.43)

The remaining part of the proof repeats the corresponding part in Theorem 2.1.2. Remark 2.1.2. The coefficient an can be calculated according to formula (2.1.37) if h  n  k and the value of the coefficient an does not depend on the parameter k. In the case where h D 1, formula (2.1.37) gives the same expressions for the coefficients an as in formula (2.1.26) in Theorem 2.1.2. The corresponding explicit expressions for the first three coefficients in the expansion (2.1.36) in terms of the coefficients that

86

2 Nonlinearly perturbed renewal equation N

appear in the perturbation condition P1.k/ if h D 2 are a2 D 

b2;0 ; b0;1

1 b3;0 b1;1 b2;0 .b3;0 C b1;1 a2 / D  C ; 2 b0;1 b0;1 b0;1  1  1 b4;0 C b2;1 a2 C b1;1 a3 C b0;2 a22 : a4 D  b0;1 2

a3 D 

(2.1.44)

2.2 Exponential expansions for improper perturbed renewal equation In this section we give exponential expansions for solutions of the improper perturbed renewal equation.

2.2.1 The asymptotically improper perturbed renewal equation with the perturbation of order O."/ Consider again, for every "  0, the renewal equation (2.1.1). We use the notations FN ."/ .t / D F ."/ .t /=F ."/ .1/; t  0; f ."/ D 1  F ."/ .1/: The following condition replaces D6 : D11 :

(a) FN ."/ ./ ) FN .0/ ./ as " ! 0, where FN .0/ .t / is a proper non-arithmetic distribution function; (b) f ."/ ! f .0/ 2 Œ0; 1/ as " ! 0.

We also assume the following Cram´er type condition (introduced in Subsection 1.4.2): C2 : There exists ı > 0 such that: R1 (a) lim0"!0 0 e ıs F ."/ .ds/ < 1; R1 (b) 0 e ıs F .0/ .ds/ > 1. As follows from Lemma 1.4.4, in the case where condition C2 (a) holds, condition D11 is implied by the following simpler condition: D12 : F ."/ ./ ) F .0/ ./ as " ! 0, where F .0/ .t / is a proper or improper distribution function not concentrated in the origin, i.e., such that F .0/ .0/ < F .0/ .1/, and FN .0/ .t / D F .0/ .t /=F .0/ .1/ is a non-arithmetic distribution function.

87

2.2 Exponential expansions for improper perturbed renewal equation

Let us again consider the characteristic equation (2.1.4). It follows from Lemma 1.4.3 that under conditions D11 and C2 there exists a unique nonnegative root ."/ of equation (2.1.4) for all " small enough. Note that ."/ D 0 if and only if f ."/ D 0 or, equivalently, ."/ > 0 if and only if f ."/ > 0. Also, ."/ ! .0/ as " ! 0. Let us introduce the following mixed moment generating functions: 

."/

Z .; r/ D

1 0

s r e s F ."/ .ds/;

r D 0; 1; : : : :

Note that, by the definition,  ."/ .; 0/ is the Laplace transform of the distribution function F ."/ .x/. Also, 1   ."/ .0; 0/ D f ."/ is the defect and  ."/ .0; r/ D m."/ r , r  1, are the corresponding moment functionals for this distribution function. Note also that  ."/ .; r/ are non-negative and non-decreasing functions in . Condition C2 implies that, for any ˇ < ı, the moment generating functions satisfy  ."/ .ˇ; r/ < 1, r  1, for " small enough. Indeed, condition C2 implies that  ."/ .ı; 0/ < 1 for " small enough, say "  "0 . Denote cr D sups0 s r e .ıˇ /s < 1. Then we have for "  "0 and r D 0; 1; : : : that  ."/ .ˇ; r/ D

Z

1 0

 cr

Z

s r e ˇs F ."/ .ds/ 1

0

e ıs F ."/ .ds/ D cr  ."/ .ı; 0/ < 1:

By condition C2 , .0/ < ı. Therefore, ˇ can be chosen such that .0/ < ˇ < ı. Also, ."/ ! .0/ as " ! 0. Thus, ."/ < ˇ < ı for all " small enough. By remarks above, we have  ."/ ..0/ ; r/;  ."/ .."/ ; r/ < 1, r D 0; 1; : : :, for all " small enough. The following perturbation condition plays a crucial role in the subsequent analysis: .k/

P3 :  ."/ ..0/ ; r/ D  .0/ ..0/ ; r/ C b1;r " C    C bkr;r "kr C o."kr / for r D 0; : : : ; k, where jbn;r j < 1, n D 1; : : : ; k  r, r D 0; : : : ; k. It is convenient to define b0;r D  .0/ ..0/ ; r/, r D 0; 1; : : : . From the definition of  it is clear that b0;0 D  .0/ ..0/ ; 0/ D 1. .k/ Note also that conditions D11 and P3 imply that .0/

 ."/ ..0/ ; r/ !  .0/ ..0/ ; r/ 2 .0; 1/ as " ! 0;

r D 0; : : : ; k:

(2.2.1) .k/

It should be noted that, in the case .0/ D 0, the perturbation condition P3 is .k/ reduced to the perturbation condition P1 . For this reason, we use the same symbols .k/ to denote the coefficients in the corresponding expansions despite that P3 involves .k/ the mixed power-exponential moments while P1 the power moments.

88

2 Nonlinearly perturbed renewal equation

We also assume that the following condition (introduced in Subsection 1.4.2) holds: F6 :

(a) limu!0 lim0"!0 supjvju jq ."/ .t C v/  q .0/ .t /j D 0 almost everywhere with respect to Lebesgue measure on Œ0; 1/; (b) lim0"!0 sup0tT jq ."/ .t /j < 1 for every T  0; P .0/ (c) limT !1 lim0"!0 h r T = h supr ht.rC1/h e . C /t jq ."/ .t /j D 0 for some h > 0 and  > 0.

We denote

R1

."/

e  s q ."/ .s/m.ds/ : R1 ."/ s F ."/ .ds/ 0 se R 1 ."/ Conditions D11 , C2 , and F6 imply that 0 e  s jq ."/ .s/jm.ds/ < 1 for all " small enough. Indeed, ."/ ! .0/ as " ! 0. Thus, ."/ < .0/ C  for all " small enough. Therefore, the renewal limit xQ ."/ .1/ is well defined for all " small enough. The following theorem generalises Theorem 2.1.2 to the case where the root .0/ of the limiting equation (2.1.4) can be equal to 0 or positive. In the first case, the theorem reduces to the case considered in Theorem 2.1.2. xQ

."/

.1/ D

0

Theorem 2.2.1. Let conditions D11 , C2 , and P3.k/ hold. Then (i) The root ."/ of equation (2.1.4) has the asymptotic expansion ."/ D .0/ C a1 " C    C ak "k C o."k /;

(2.2.2)

where the coefficients an are given by the recurrence formulas a1 D b1;0 =b0;1 and, in general, for n D 1; : : : ; k,  n1 X 1 bnq;1 aq bn;0 C an D b0;1 qD1

C

X

n X

2mn qDm

bnq;m 

X

q1 Y

 ; (2.2.3)

n app =np Š

n1 ;:::;nq1 2Dm;q pD1

where Dm;q , for every 2  m  q < 1, is the set of all nonnegative, integer solutions of the system n1 C    C nq1 D m; n1 C    C .q  1/nq1 D q:

(2.2.4)

(ii) If bl;0 D 0, l D 1; : : : ; r, for some 1  r  k, then a1 ; : : : ; ar D 0. If bl;0 D 0, l D 1; : : : ; r  1 but br;0 < 0, for some 1  r  k, then a1 ; : : : ; ar1 D 0 but ar > 0.

2.2 Exponential expansions for improper perturbed renewal equation

89

.r/

(iii) If, additionally, conditions B3 , for some 1  r  k, and F6 hold, then the following asymptotic relation holds: x ."/ .t ."/ / expf..0/ C a1 " C    C ar1 "r1 /t ."/ g ! e r ar xQ .0/ .1/ as " ! 0:

(2.2.5)

Before we prove Theorem 2.2.1 we make a few remarks. Remark 2.2.1. Formula (2.2.3) has the same form as formula (2.1.26). The only difference is that, in (2.2.3), the coefficients bl;r are taken from perturbation expansions .k/ for the mixed power-exponential moments  ."/ ..0/ ; r/ given in condition P3 while, in (2.1.26), the coefficients bl;r are taken from perturbation expansions for the mixed ."/ .k/ power moments mr given in condition P1 . In particular, a1 D 

b1;0 ; b0;1

1 b2;0 b1;1 b1;0 .b2;0 C b1;1 a1 C b0;2 a12 / D  C  2 b0;1 2 b0;1 b0;1 1  1 b3;0 C b2;1 a1 C b1;1 a2 C b1;2 a12 C b0;2 a1 a2 C a3 D  b0;1 2

a2 D 

1

2 b0;2 b1;0 3 2b0;1

;

 1 b0;3 a13 : (2.2.6) 6

Remark 2.2.2. The the case where F ."/ D F .0/ and q ."/ D q .0/ do not depend on " is also covered by Theorem 2.2.1. In this case, condition P3.k/ holds for every k  1 with the coefficients satisfying bl;r D 0 for r  0, l  1. Hence, the expansion (2.2.2) takes place for every k  1 with the coefficients al D 0 for l D 1; : : : ; k. In this case, the expansion (2.2.2) takes the trivial form .0/ D .0/ C o."k / which implies o."k /  0 and it does not give any information. However, the statement (iii) yields the well-known (Feller 1966) exponential asymptotics for solutions of the renewal equation under the Cram´er type condition for the distribution F .0/ . The transformation of conditions C2 and F6 , in this case, is obvious. One can always take t ."/ D "1 , so the condition that balances the rate of perturbation and the rate of growth of time will automatically be satisfied (with r D 1). The statement (iii) then takes the form .0/ 1 x .0/ ."1 /=e  " ! xQ .0/ .1/ as " ! 0. Since 0 < " ! 0 in an arbitrary way, the .0/ last relation is equivalent to the relation x .0/ .t /=e  t ! xQ .0/ .1/ as t ! 1. Proof of Theorem 2.2.1. By Lemma 1.4.3, ."/ D ."/  .0/ ! 0 as " ! 0:

(2.2.7)

90

2 Nonlinearly perturbed renewal equation

The Taylor expansion for the function e s yields e s

."/

D e s

.0/



1 C s."/ =1Š C    C s k .."/ /k =kŠ C s kC1 .."/ /kC1 e sj

."/ j

."/ kC1 .s/=.k C 1/Š ;

."/

(2.2.8)

."/

where kC1 .s/, s  0, is a continuous function such that 0  kC1 .s/  1; s  0. Recall that .0/ < ı and ."/ ! 0. Therefore, there exist ˇ < ı and "1 D "1 .ˇ/ such that ."/  .0/ C j."/ j < ˇ for " small enough, say "  "1 . By the remarks above, one can also assume that there exists "2 D "2 .ˇ/ such that  ."/ .ˇ; r/ < 1, r D 0; 1; : : :, for "  "2 . By condition C2 , Z 1 Z 1 kC1 ˇs ."/ lim s e F .ds/  lim ckC1 e ıs F ."/ .ds/ < 1; (2.2.9) 0"!0 0

0"!0

0

where ckC1 D sups0 s kC1 e .ıˇ /s < 1. It follows from relation .2:2:9/ that there exists "3 D "3 .ˇ/ such that, for "  "3 , Z 1 1 sup s kC1 e sˇ F ."/ .ds/ < 1: (2.2.10) MkC1 D .k C 1/Š ""3 0 Let us define "4 D "4 .ˇ/ D min."1 .ˇ/; "2 .ˇ/; "3 .ˇ//:

(2.2.11)

Integrating (2.2.8) and taking condition C2 into account one obtains for "  "4 that Z 1 ."/ e s F ."/ .ds/ 1D 0

D  ."/ ..0/ ; 0/ C  ."/ ..0/ ; 1/."/ =1Š C    ."/

C  ."/ ..0/ ; k/.."/ /k =kŠ C .."/ /kC1 MkC1 kC1 ; where

R1 ."/ kC1

D

."/

(2.2.12)

."/

s kC1 e s kC1 .s/F ."/ .ds/ R1 2 Œ0; 1: sup""3 0 s kC1 e sˇ F ."/ .ds/ 0

Formula (2.2.12) can be rewritten in the form  ."/ ..0/ ; 1/."/ =1Š C  ."/ ..0/ ; 2/.."/ /2 =2Š C    ."/

C  ."/ ..0/ ; k/.."/ /k =kŠ C .."/ /kC1 MkC1 kC1 D 1   ."/ ..0/ ; 0/:

(2.2.13)

2.2 Exponential expansions for improper perturbed renewal equation

91

The following proof is analogous to that given in Theorem 2.1.2 with the only change that the quantity ."/ is replaced with ."/ . The difference ."/ ! 0 and the sum of all terms in the left-hand side in (2.2.13), starting with the second one, is o.."/ /. Also,  ."/ ..0/ ; r/ !  .0/ ..0/ ; r/ 2 .0; 1/ as " ! 0 for r D 0; : : : ; k. Recall also that b0;0 D 1 and, therefore, the expression in the right-hand side is of order ". Dividing both sides in (2.2.13) by b0;1 " and evaluating .k/ the corresponding limits we obtain using P3 that ."/ =" ! b1;0 =b0;1 . This means that ."/ D ."/  .0/ can be represented as ."/

."/ D ."/  .0/ D a1 " C 1 ;

(2.2.14)

."/

where a1 D b1;0 =b0;1 and 1 D o."/. Relation (2.2.14) reduces to (2.2.2) in the case k D 1. .k/ Let k  2. Substituting the expansions given in condition P3 and (2.2.14) into (2.2.13) and using the identity a1 D b1;0 =b0;1 we get ."/

.b1;1 " C o."//a1 " C .b0;1 C b1;1 " C o."//1 ."/

C .b0;2 C o.1//.a1 " C 1 /2 =2Š C    k 2 2 C .b0;k C o.1//.a1 " C ."/ 1 / =kŠ D  b2;0 " C o." /:

(2.2.15)

Dividing both sides in (2.2.15) by b0;1 "2 and evaluating the corresponding limits we ."/ obtain that b1;1 a1 =b0;1 C 1 ="2 C b0;2 a12 =2b0;1 ! b2;0 =b0;1 . This is equivalent to the following asymptotic relation: ."/ 2 ."/ 1 D a2 " C 2 ;

(2.2.16)

."/

1 where a2 D b0;1 .b2;0  b1;1 a1  12 b0;2 a12 / and 2 D o."2 /. Relations (2.2.14) and (2.2.16) yield relation (2.2.2) for k D 2. The expression for a2 given above coincides with formula (2.2.3) for k D 2. Repeating the above argument we obtain expansion (2.2.2) and formula (2.2.3) for k > 2. However, formula (2.2.3) can be obtained in a simpler way when the asymptotic expansion (2.2.2) has already been proved. From (2.2.13) we get the following formal equation:

.b0;1 C b1;1 " C    /.a1 " C a2 "2 C    /=1Š C .b0;2 C b1;2 " C    /.a1 " C a2 "2 C    /2 =2Š C    D .b1;0 " C b2;0 "2 C    /:

(2.2.17)

Equating the coefficients of "n for n  1 in the right and left-hand sides of (2.2.17) we obtain formula (2.2.3) for calculating the coefficients a1 ; : : : ; ak .

92

2 Nonlinearly perturbed renewal equation

The first summand in (2.2.3) reflects the contribution of the sum in the right-hand side of (2.2.17). The second term in the sum in (2.2.3) gives a contribution of the first product of the sums in the left-hand side of (2.2.17). The sum with index 2  m  n in (2.2.3) reflects the contribution of the product of the sums .b0;m C   /.a1 "C   /m =mŠ in the right-hand side of (2.2.17). In this case, we collect all terms in this product such that the sum .b0;m C    / contributes "nq and .a1 " C    /m =mŠ the "q for m  q  n. Note that the cases 1  q < m are impossible, since the minimal power of ", which .a1 " C    /m =mŠ can contribute to this product, is m. The coefficient of n "q , which comes from .a1 " C    /m =mŠ, is the sum of the products a1n1    aq q , where the nonnegative integer coefficients n1 ; : : : ; nq must satisfy the obvious relations n1 C    C nq D m and n1 C    C q nq D q. Only the values 0 and 1 are suitable for the coefficient nq . Indeed, if nq D 1, then all other coefficients n1 ; : : : ; nq1 must equal 0. In this case, n1 C    C nq D 1, which contradicts the restriction 2  m  n. Therefore, the coefficient nq must be 0. Finally, combinatoric counting n of the coefficients in the product a1n1    aq q yields the number mŠ=n1 Š    nq Š. Also, nq D 0, as was mentioned above. The coefficient mŠ cancels, since it also appears as the denominator of .a1 " C    /m =mŠ. This completes the proof of (i). Let us use now Theorem 1.4.2. This theorem implies that under conditions D11 , C3 and F6 , for any 0  t ."/ ! 1, the following asymptotical relation holds: x ."/ .t ."/ / ! xQ .0/ .1/ as " ! 0: ."/ ."/ expf t g

(2.2.18)

Let us prove (ii). Under the balancing condition "r t ."/ ! r , the asymptotics relation (2.2.18) can be rewritten in the following equivalent form: xQ ."/ .t ."/ / D

x ."/ .t ."/ /

e ..0/ Ca1 "CCar "r Co."r //t ."/ x ."/ .t ."/ /  ! xQ .0/ .1/ as " ! 0: e ..0/ Ca1 "CCar "r /t ."/

(2.2.19)

The latter asymptotic relation is equivalent to (2.2.5). Remark 2.2.3. The asymptotic relation (2.2.19) can be written assuming the weaker balancing condition lim0 0, r D 1; : : : ; NQ . .kN / .k/ Condition P4 is more general than condition P3 , and the first one reduces to the second one if h D 1, k0 D k (in this case, NQ D k) and kr D k  r, r D 1; : : : ; NQ . The following theorem generalises Theorems 2.1.2. The proof is analogous to the one given in Theorems 2.1.3 and 2.2.1. .kN /

Theorem 2.2.2. Let conditions D11 , C2 , and P4 hold. Then (i) For all " small enough, equation (2.1.4) has a unique nonnegative root ."/ that has the following asymptotic expansion: ."/ D .0/ C ah "h C    C ak "k C o."k /;

(2.2.20)

where k D min.k0 ; k1 Ch; : : : ; kNQ ChNQ / and the coefficients an , n D h; : : : ; k, are given by the recurrence formulas ah D bh;0 =b0;1 and, in general, for n D h; : : : ; k, we have  n1 X 1 bn;0 C an D b0;1 bni;1 ai iDh

C

Œn= Xh

n X

r D2 j Dr h

bnj;r 

X

jY 1

 aini =ni Š ;

(2.2.21)

nh ;:::;nj 1 2Dh;r;j iDh

where Dh;r;j , for every h  1 and 2  r  j < 1, is the set of all nonnegative, integer solutions of the system nh C    C nj 1 D r; hnh C    C .j  1/nj 1 D j:

(2.2.22)

(ii) If bl;0 D 0, l D h; : : : ; r, for some h  r  k, then a1 ; : : : ; ar D 0. If bl;0 D 0, l D h; : : : ; r  1 but br;0 < 0 for some h  r  k, then ah ; : : : ; ar1 D 0 but ar > 0.

94

2 Nonlinearly perturbed renewal equation .r/

(iii) If, additionally, conditions B3 , for some h  r  k, and F6 hold, then the following asymptotic relation holds: x ."/ .t ."/ / expf..0/ C ah "h C    C ar1 "r1 /t ."/ g ! e r ar xQ .0/ .1/ as " ! 0:

(2.2.23)

Remark 2.2.4. Formula (2.2.21) has the same form as formula (2.1.37). The only difference is that, in (2.2.21), the coefficients bl;r are taken from the perturbation expansions for the mixed power-exponential moments  ."/ ..0/ ; r/ given in condition N P4.k/ while, in (2.1.37), the coefficients bl;r are taken from the perturbation expansions ."/ .kN / for the mixed power moments mr given in condition P2 . In the case where h D 1, we have the following formulas for the first three coefficients: a1 D 

b1;0 ; b0;1 1

1 b2;0 b1;1 b1;0 .b2;0 C b1;1 a1 C b0;2 a12 / D  C  a2 D  2 b0;1 2 b0;1 b0;1 1  1 b3;0 C b2;1 a1 C b1;1 a2 C b1;2 a12 C b0;2 a1 a2 C a3 D  b0;1 2

2 b0;2 b1;0 3 2b0;1

;

 1 b0;3 a13 : (2.2.24) 6

In the case where h D 2, we have the following formulas for the first three coefficients: a2 D 

b2;0 ; b0;1

b3;0 b1;1 b2;0 C ; 2 b0;1 b0;1 b0;1  1  1 b4;0 C b2;1 a2 C b1;1 a3 C b0;2 a22 : a4 D  b0;1 2

a3 D 

1

.b3;0 C b1;1 a2 / D 

(2.2.25)

2.3 Asymptotic expansions for renewal limits In this section we give asymptotic expansions for renewal limits for proper and improper perturbed renewal equations.

2.3.1 Asymptotic expansions for renewal limits for proper perturbed renewal equations First we consider the simplest case where (a) F ."/ .C1/ D 1 for all "  0.

95

2.3 Asymptotic expansions for renewal limits

Formula (1.2.35) gives, in this case, an expression for the renewal limit (see, Subsection 1.2.6), x ."/ .1/ D

."/

q0

."/

(2.3.1)

;

m1 where ."/

Z

q0 D

1 0

q ."/ .s/m.ds/;

."/

m1 D

Z

1 0

sF ."/ .ds/:

."/

Convergence of the renewal limits x .1/ requires a perturbation condition sim.k/ ."/ pler than P1 and involving only the moment functionals m1 , .k/

."/

.0/

.0/

P5 : m1 D m1 C b1;1 " C    C bk;1 "k C o."k /, where m1 2 .0; 1/ and jbn;1 j < 1, n D 1; : : : ; k. It convenient to also define b0;1 D m.0/ 1 . Also the following perturbation condition imposed on the moment functionals for forcing functions is required: .k/

."/

.0/

.0/

P6 : q0 D q0 C c1;0 " C    C ck;0 "k C o."k /, where jq0 j < 1 and jcn;0 j < 1, n D 1; : : : ; k. We define c0;0 D q0.0/ . .0/

.0/

Note that m1 and q0 in the condition above are just some finite constants that R1 R1 .0/ .0/ may be or not of the form m1 D 0 sF .0/ .ds/ and q0 D 0 q .0/ .s/m.ds/. The following theorem improves Theorem 1.2.4. .k/

.k/

Theorem 2.3.1. Let (a) F ."/ .C1/ D 1 for all "  0 and conditions P5 and P6 hold. Then the functional x ."/ .1/ has the asymptotic expansions: x

."/

.1/ D

q0.0/ C c1;0 " C    C ck;0 "k C o."k / .0/

m1 C b1;1 " C    C bk;1 "k C o."k /

D x .0/ .1/ C f1 " C    C fk "k C o."k /;

(2.3.2)

where the coefficients fn are given by the recurrence formulas f0 D x .0/ .1/ D c0;0 =b0;1 , and in general for n D 0; : : : ; k,   n1 X 1 cn;0  bnq;1 fq : fn D b0;1 qD0

(2.3.3)

96

2 Nonlinearly perturbed renewal equation

Proof. Using formula (2.3.1) we can write the rational expansion (2.3.2). It always (see, Lemma 8.1.1) can be transformed to a polynomial form, since b0;1 ¤ 0. Hence, we can write the following: .0/

.x .0/ .1/ C f1 " C    C fk "k C o."k //.m1 C b1;1 " C    C bk;1 "k C o."k // D c0;0 C c1;0 " C    C ck;0 "k C o."k /:

(2.3.4)

The recurrence formulas (2.3.3) can be obtained by grouping the coefficients of "n in equation (2.3.4). Remark 2.3.1. The coefficients fn , n D 0; : : : ; k, can be calculated according to formula (2.3.4) and the value of the coefficient fn does not depend on the parameter k. In particular, the formulas for first three coefficients take the following form: f0 D

c0;0 ; b0;1

f1 D

1 b0;1

.c1;0  b1;1 f0 / D

c1;0 c0;0 b1;1  ; 2 b0;1 b0;1

1 .c2;0  b2;1 f0  b1;1 f1 / b0;1

f2 D

2 c0;0 b1;1 c2;0 c0;0 b2;1 C c1;0 b1;1  C : 2 3 b0;1 b0;1 b0;1

D

(2.3.5)

2.3.2 Asymptotic expansions for renewal limits for improper perturbed renewal equations Let us now consider the general case where (b) F ."/ .C1/  1 for all "  0. The following formula, which repeats (1.4.39), gives, in this case, an expression for the renewal limit, xQ ."/ .1/ D where  ."/ .; r/ D

Z

1 0

! ."/ .."/ ; 0/ ;  ."/ .."/ ; 1/

s r e s F ."/ .ds/;

  0;

(2.3.6)

r D 0; 1; : : : :

Let us now define for the mixed power-exponential moment functionals for the forcing functions, the following: Z 1 ! ."/ .; r/ D s r e s q ."/ .s/m.ds/;   0; r D 0; 1; : : : ; 0

2.3 Asymptotic expansions for renewal limits

and !N ."/ .; r/ D

Z

1 0

s r e s jq ."/ .s/jm.ds/;

  0;

97

r D 0; 1; : : : :

Condition F6 implies that, for any 0 < ˇ < .0/ C  , the moment generating functions satisfy !N ."/ .ˇ; r/ < 1, r D 0; 1; : : :, for all " small enough. Indeed, condition F6 implies that !N ."/ ..0/ C ; 0/ < 1 for " small enough, say .0/ "  "0 . Denote cr D sups0 s r e . C ˇ /s < 1. Then, we have for "  "0 and r D 0; 1; : : : that Z 1 ."/ !N .ˇ; r/ D s r e ˇs jq ."/ .s/jm.ds/ 0 Z 1 .0/ e . C /s jq ."/ .s/jm.ds/ D cr !N ."/ ..0/ C ; 0/ < 1:  cr 0

."/

Note that !N .; r/ are non-negative and non-decreasing functions of   0. Also, < .0/ C  , since ."/ ! .0/ as " ! 0. Thus, !N ."/ .."/ ; r/, !N ."/ ..0/ ; r/ < 1, r D 0; 1; : : :, for all " small enough. Therefore, the renewal limit xQ ."/ .1/ is well defined for all " small enough. Let us now formulate perturbation conditions for mixed power-exponential moment functionals for the forcing functions, ."/

P7.k/ : ! ."/ ..0/ ; r/ D ! .0/ ..0/ ; r/ C c1;r " C    C ckr;r "kr C o."kr / for r D 0; : : : ; k, where jcn;r j < 1, n D 1; : : : ; k  r, r D 0; : : : ; k. It is convenient to set c0;r D ! .0/ ..0/ ; r/, r D 0; 1; : : : . The following theorem improves Theorem 1.4.4. .kC1/

.k/

, and P7 Theorem 2.3.2. Let conditions D11 , C2 , F6 , P3 ."/ tional xQ .1/ has the following asymptotic expansions: xQ ."/ .1/ D

hold. Then the func-

! .0/ ..0/ ; 0/ C f10 " C    C fk0 "k C o."k /

 .0/ ..0/ ; 1/ C f100 " C    C fk00 "k C o."k /

D xQ .0/ .1/ C f1 " C    C fk "k C o."k /;

(2.3.7)

where the coefficients fn0 ; fn00 are given by the formulas f00 D ! .0/ ..0/ ; 0/ D c0;0 , f10 D c1;0 C c0;1 a1 , f000 D  .0/ ..0/ ; 1/ D b0;1 , f100 D b1;1 C b0;2 a1 , and in general for n D 0; : : : ; k, fn0 D cn;0 C

n X

cnq;1 aq

qD1

C

X

n X

2mn qDm

cnq;m 

X

q1 Y

n1 ;:::;nq1 2Dm;q pD1

n

app =np Š;

(2.3.8)

98

2 Nonlinearly perturbed renewal equation

and fn00

D bn;1 C

n X

bnq;2 aq

qD1

X

C

n X

X

bnq;mC1 

2mn qDm

q1 Y

n

app =np Š;

(2.3.9)

n1 ;:::;nq1 2Dm;q pD1

and the coefficients fn are given by the recurrence formulas f0 D xQ .0/ .1/ D f00 =f000 and in general for n D 0; : : : ; k,   n1 X 0 00 fn D fn  fnq fq =f000 : (2.3.10) qD0

Proof. We can write the asymptotic expansion, in powers of the difference ."/ D ."/  .0/ , for the numerator and the denominator in the quotient expression for xQ ."/ .1/ given by (2.3.6). Recall that .0/ < ı, where ı is defined in C2 , and ."/ ! 0. Hence, for any .0/ < ˇ < ı there exists "1 D "1 .ˇ/ such that ."/  .0/ C j."/ j  ˇ for "  "1 . By the remarks above, there exists "2 D "2 .ˇ/ > 0 such that ! ."/ .ˇ; r/ < 1, r D 0; 1; : : :, and  ."/ .ˇ; r/ < 1, r D 0; 1; : : :, for "  "2 . Let us use ones more the Taylor expansion for the function e s , ."/ .0/ e s D e s 1 C s."/ =1Š C    C s k .."/ /k =kŠ

."/ ."/ (2.3.11) C s kC1 .."/ /kC1 e sj j kC1 .s/=.k C 1/Š ; ."/

."/

where kC1 .s/; s  0, is a continuous function such that 0  kC1 .s/  1, s  0. By condition C2 , Z 1 Z 1 kC2 ˇs ."/ lim s e F .ds/  lim ckC2 e ıs F ."/ .ds/ < 1; (2.3.12) 0"!0 0

0"!0

0

and, by condition F6 , Z 1 lim s kC1 e ˇs jq ."/ .s/jm.ds/ 0"!0 0

Z  lim ckC1 0"!0

1 0

e ıs jq ."/ .s/jm.ds/ < 1;

(2.3.13)

where ck D sups0 s k e .ıˇ /s < 1. It follows from relations .2:2:9/ and .2:3:13/ that there exists "3 D "3 .ˇ/ > 0 such that, for "  "3 , Z 1 1 sup s kC2 e sˇ F ."/ .ds/ < 1; (2.3.14) MkC2 D .k C 1/Š ""3 0

2.3 Asymptotic expansions for renewal limits

99

and MO kC1 D

1 sup .k C 1/Š ""3

Z

1 0

s kC1 e sˇ jq ."/ .s/jm.ds/ C

1 < 1: .k C 1/Š

(2.3.15)

Define "4 D "4 .ˇ/ D min."1 .ˇ/; "2 .ˇ/; "3 .ˇ//:

(2.3.16)

Substituting the Taylor expansion (2.3.11) into the integrals that define  ."/ .."/ ; 1/ and ! ."/ .."/ ; 0/, we get for "  "4 that  ."/ .."/ ; 1/ D  ."/ ..0/ ; 1/ C  ."/ ..0/ ; 2/."/ C    ."/ C  ."/ ..0/ ; k C 1/.."/ /k =kŠ C .."/ /kC1 MkC2 kC2 ; (2.3.17)

where

R1

."/ kC2

."/

."/

s kC2 e s kC2 .s/F ."/ .ds/ R1 D 2 Œ0; 1; sup""3 0 s kC2 e sˇ F ."/ .ds/ 0

and ! ."/ .."/ ; 0/ D ! ."/ ..0/ ; 0/ C ! ."/ ..0/ ; 1/."/ C    ."/ C ! ."/ ..0/ ; k/.."/ /k =kŠ C .."/ /kC1 MO kC1 OkC1 ;

where

R1

."/ OkC1 D

(2.3.18)

."/ s kC1 e s kC1 .s/q ."/ .s/m.ds/ R1 2 Œ1; 1: sup""3 0 s kC1 e sˇ jq ."/ .s/jm.ds/ C 1 ."/

0

It was proved in Theorem 2.2.1 that the functions ."/ D ."/  .0/ can be represented in the form of the asymptotic expansion ."/ D a1 " C    C ak "k C o."k /. The functions ! ."/ ..0/ ; n/ and  ."/ ..0/ ; n/ can also be represented in the form of asymptotic expansions due to conditions P3.kC1/ and P7.k/ , respectively. Substituting these expansions into (2.3.17) and (2.3.18) we obtain ! ."/ .."/ ; 0/ D .c0;0 C    C ck;0 "k C o."k // C .c0;1 C    C ck1;1 "k1 C o."k1 //.a1 " C    C ak "k C o."k //=1Š C    C .c0;k C o.1//.a1 " C    C ak "k C o."k //k =kŠ C o."k /; (2.3.19)

100

2 Nonlinearly perturbed renewal equation

and  ."/ ..0/ ; 1/ D .b0;1 C    C bk;1 "k C o."k / C .b0;2 C    C bk1;2 "k1 C o."k1 //.a1 " C    C ak "k C o."k //=1Š C    C .b0;kC1 C o.1//.a1 " C    C ak "k C o."k //k =kŠ C o."k /:

(2.3.20)

By grouping the coefficients of powers of " in these expansions one can transform them to the forms described in (2.3.7)–(2.3.9), i.e., ! ."/ .."/ ; 0/ D f00 C    C fk0 "k C o."k /;

(2.3.21)

 ."/ .."/ ; 1/ D f000 C    C fk00 "k C o."k /:

(2.3.22)

and

The transition of the expansion (2.3.7) to a polynomial form is obvious (see, Lemma 8.1.1), since f000 ¤ 0. The corresponding coefficients can be found by grouping the coefficients of "n in the equation .f0 C    C fk "k C o."k //.f000 C    C fk00 "k C o."k // D f00 C    C fk0 "k C o."k /:

(2.3.23)

The relations (2.3.10) can be found by grouping the coefficients of "n in equation (2.3.23). It is useful to note that in the case where f .0/ D 0, the characteristic root .0/ D 0 and  ."/ .0; r/ D m."/ r , r  1, reduce to the usual power moments for the distributions ."/ functions F .x/, and ! ."/ .0; r/ D qr."/ , r  1, reduce to the corresponding power moment functionals for the forcing functions q ."/ .s/, Z 1 ."/ qr D s r q ."/ .s/m.ds/; r  1: 0

.kC1/

reduces to the perturbation condition In this case, the perturbation condition P3 .kC1/ P1 . .k/ The perturbation condition P7 also reduces, under condition F6 , to the following simpler form: P8.k/ : qr."/ D qr.0/ C c1;r " C    C ckr;r "kr C o."kr / for r D 0; : : : ; k, where jcn;r j < 1, n D 1; : : : ; k  r, r D 0; : : : ; k.

2.3 Asymptotic expansions for renewal limits

101

Remark 2.3.2. The following formulas can be obtained for the first three coefficients given by formulas (2.3.8), (2.3.9), and (2.3.10): f00 D c0;0 ; f10 D c1;0 C c0;1 a1 D c1;0 

c0;1 b1;0 ; b0;1

f20 D c2;0 C c0;1 a2 C c1;1 a1 C

c0;2 a12 ; 2

f000 D b0;1 ; f100 D b1;1 C b0;2 a1 D b1;1 

b0;2 b1;0 ; b0;1

f200 D b2;1 C b0;2 a2 C b1;2 a1 C f0 D

f00 c0;0 D ; f000 b0;1

f1 D

1 .f 0  f0 f100 / f000 1

D f2 D

b0;3 a12 ; 2

c1;0 c0;1 b1;0 c0;0 b1;1 c0;0 b0;2 b1;0   C ; 2 2 3 b0;1 b0;1 b0;1 b0;1 1 .f 0  f0 f200  f1 f100 /: f000 2

(2.3.24)

Chapter 3

Nonlinearly perturbed regenerative processes

Chapter 3 contains results concerning mixed ergodic and limit theorems and mixed ergodic and large deviation theorems for nonlinearly perturbed regenerative processes. These theorems are given in a form of exponential asymptotics for joint distributions of positions of regenerative processes and regenerative stopping times. These distributions satisfy some renewal equations and the corresponding theorems are obtained by applying results of Chapters 1 and 2 to these renewal equations. ."/ We consider a regenerative process ."/ .t /, t  0, with regeneration times n , ."/ ."/ and a regenerative stopping time  that regenerates jointly with the process .t /, t  0, at times n."/ . Both the regenerative process ."/ .t /, t  0, and the regenerative stopping time ."/ are assumed to depend on a small perturbation parameter "  0. The processes ."/ .t /, t  0, for " > 0 are considered as a perturbation of the process .0/ .t /, t  0, and therefore we assume some weak continuity conditions for certain characteristic quantities of these processes regarded as functions of " at point " D 0. As far as the regenerative stopping times are concerned, we consider two cases. The first one is where the random variables ."/ are stochastically unbounded, i.e., ."/ tend to 1 in probability as " ! 0. The second one is where the random variables ."/ are stochastically bounded as " ! 0. The object of our study is the joint distributions Pf ."/ .t / 2 A; ."/ > t g and their asymptotic behaviour as t ! 1 and " ! 0. We also present results concerning conditions for convergence and finding asymptotic expansions for quasi-stationary distributions of regenerative processes. Two types of characteristic quantities that are related to one regeneration period play a crucial role in our analysis. These are the stopping probability f ."/ D Pf."/ < ."/ n ."/  ."/ /

1."/ g and the power moments of the regeneration times m."/ n D E. 1 / . 1 ."/

."/

or their mixed power-exponential analogues  ."/ .; n/ D E. 1 /n e 1 .."/  ."/

1 / up to the order n D k. One of the main new elements in this study is that we consider a model with nonlinear perturbations. This means that the stopping probabil."/ ities f ."/ and the moments m."/ n or  .; n/ are nonlinear functions of ". However, we restrict our attention to the case where the nonlinear perturbations are smooth and assume that these functions can be expanded in a power series with respect to " up to and including the order k.

104

3 Nonlinearly perturbed regenerative processes

It turns out that the relationship between the rates with which " tends to zero and the time t tends to infinity is important for the results. Without loss of generality we assume that the time t D t ."/ is a function of the parameter ". The balance between the rate of perturbation and the rate of growth of time is characterised by the following condition controlled by an integer parameter r, 1  r  k: "r t ."/ ! r < 1 as " ! 0:

(3.0.1)

Under the above assumptions and some natural additional conditions of Cram´er type for regeneration times, we obtain the asymptotic relation Pf ."/ .t ."/ / 2 A; ."/ > t ."/ g !  .0/ .A/e r ar as " ! 0: expf.0 C a1 " C    C ar 1 "r 1 /t ."/ g

(3.0.2)

This asymptotic relation is provided with an explicit expression for  .0/ .A/, and an explicit recurrence algorithm for calculating the coefficients a1 ; : : : ; ar as rational functions of the coefficients in the expansions for the stopping probabilities f ."/ and ."/ the moments mn , n D 1; : : : ; k, or  ."/ .; n/, n D 1; : : : ; k. The case 0 D 0 corresponds to a model with stochastically unbounded random variables ."/ , while the case 0 > 0 corresponds to a model with stochastically bounded random variables ."/ . The asymptotic relation (3.0.2) describes in these cases pseudo-stationary and quasi-stationary phenomena for perturbed regenerative processes, respectively. To clarify the meaning of the asymptotic relation (3.0.2) for different k, let us consider the case where 0 D 0. If r D k D 1, relation (3.0.1) is reduced to "t ."/ ! 1 , and the asymptotic relation (3.0.2) can be rewritten in the form Pf ."/ .t ."/ / 2 A; ."/ > t ."/ g !  .0/ .A/e 1 a1 as " ! 0:

(3.0.3)

Asymptotic relation (3.0.3) shows that the position of the regenerative process .t ."/ / and the normalised regenerative stopping time "."/ are asymptotically independent and have, in the limit, a stationary distribution and an exponential distribution, respectively. This can be interpreted as a mixed ergodic theorem (for the regenerative processes) and a limit theorem (for regenerative stopping times). The asymptotic relation (3.0.3) describes the simplest form of pseudo-stationary phenomena for perturbed regenerative processes. In the literature, the term transient is also used in connection with such a kind of asymptotic relations. The simplest form of pseudo-stationary asymptotics given by (3.0.3), as well as the corresponding results on rates of convergence do not give satisfactory information about the asymptotic behaviour of the joint distribution of the normalised regenerative stopping times and the positions of the regenerative process in zones of large deviations for "t ."/ ! 1. In particular, they do not shed light upon the asymptotic behaviour

3

Nonlinearly perturbed regenerative processes

105

of relative errors of approximation in large deviation zones, which is of great interest. Relation (3.0.2), for the case k > 1, provides a means to describe this asymptotics. In the case r D k D 2, the balancing relation (3.0.1) is reduced to "2 t ."/ ! 2 and the asymptotic relation (3.0.2) can be rewritten in the form Pf ."/ .t ."/ / 2 A; ."/ > t ."/ g ! e 2 a2 as " ! 0:  .0/ .A/ expfa1 "t ."/ g

(3.0.4)

The limit in the right-hand side gives information about the asymptotic behaviour of the relative error of the approximation in (3.0.4) in the large deviation zone of the orders "t ."/ D o."1 / or "t ."/ D O."1 /. If 2 D 0, then "t ."/ D o."1 / and the asymptotic relative error is 0. Note that this case also covers the situation where "t ."/ is bounded, which corresponds to the asymptotic relation (3.0.3) with r D k D 1. This is already an extension of the asymptotic result since it is possible that "t ."/ ! 1. If 2 > 0, then "t ."/ D O."1 /, and the asymptotic relative error is 1  e 2 a2 . It differs from 0. Therefore, o."1 / is an asymptotic bound for the large deviation zone with the asymptotic relative error 0. Relation (3.0.3) can be interpreted as a new kind of mixed ergodic and large deviation theorems for nonlinearly perturbed regenerative processes. A similar interpretation can be made of the asymptotic relation (3.0.2) for k > 2. As in the linear case, relation (3.0.2) can also be interpreted as a mixed limit and ergodic theorem if r D 1. On the other hand, if 1 < r  k this relation is naturally considered as a mixed large deviation theorem (for stopping times) and an ergodic theorem (for regenerative processes), since "r t ."/ ! 1 in this case. Asymptotics of the type (3.0.2) appear in the literature mainly for the case k D 1, which corresponds to a model of linearly perturbed processes. Our main result is the asymptotics (3.0.2) for k > 1, which corresponds to a model of nonlinearly perturbed processes. In Theorems 3.2.1–3.3.6 we give an exponential asymptotics which describes pseudo-stationary and quasi-stationary phenomena in perturbed regenerative processes. We improve this asymptotics to the form of exponential expansions in mixed ergodic and large deviation and ergodic theorems for nonlinearly perturbed regenerative processes and regenerative stopping times. This is done in Theorems 3.4.2 and 3.4.4. Finally, we give asymptotic expansions for stationary and quasi-stationary distributions of nonlinearly perturbed regenerative processes in Theorems 3.5.1, 3.5.2, and 3.5.4, 3.5.5. The asymptotics of type (3.0.2) makes a basis for studies of the so-called mixed ergodic and limit theorems and mixed ergodic and large deviation theorems for perturbed semi-Markov processes with absorption, which is done in Chapters 4–5.

106

3 Nonlinearly perturbed regenerative processes

3.1 Regenerative processes and regenerative stopping times In this section we introduce an object that plays a key role in our studies. These are objects are regenerative processes and regenerative stopping moments.

3.1.1 Regenerative processes and regenerative stopping times Let consider a stochastic process .t /, t  0, with a measurable phase space X (with the corresponding -algebra  of measurable subsets) defined on some probability space .; F ; P /. We assume that the process .t /; t  0, is measurable in the sense that .t; !/, .t; !/ 2 Œ0; 1/   are measurable functions of .t; !/. Let also 0 D 0  1  : : : be a monotonically non-increasing sequence of nonnegative random variables defined on the same probability space .; F ; P /. .n/ We denote by .n/ .t / D . n C t /, t  0, and k D nCk  n , k D 0; 1; : : :, for every n D 0; 1; : : :, the shifted to time n versions of the process .t / and the sequence

k . Denote by F ./ n the -algebra of random events from F generated by the random variables .s/, 0  s < n (i.e., by random variables . n  s ^ n /; s > 0), and k , k D 0; : : : ; n. This -algebra includes random events determined by the trajectory of the process .t / before the moment n and the moments k for k  n. We also denote by F n.C/ the -algebra of random events from F generated by the random variables .n/ .t /, t  0, and k.n/ , k D 0; 1; : : : . This -algebra includes the random events determined by the trajectory of the process .t / after and including the moment n and .n/ the shifted moments k D nCk  n , k D 0; 1; : : : . The stochastic process .t /, t  0, is a regenerative process with regeneration times

n , n D 0; 1; : : :, if it possess the following two properties: ./

R1 : The -algebras Fn

.C/

and Fn

are independent for every n D 0; 1; : : :;

and .n/

R2 : The joint finite dimensional distributions Pf .n/ .tr / 2 Ar ; r  sr ; r D 1; : : : ; mg D Pf .tr / 2 Ar ; r  sr ; r D 1; : : : ; mg, Ar 2 , tr ; sr  0, r D 1; : : : ; m, m  1 are the same for every n D 0; 1; : : : . According to conditions R1 and R2 , for every n D 0; 1; : : :, we have .n/ .t / D .n/ . n C t /, t  0, and k D nCk  n , k D 0; 1; : : :, is a probability copy of the regenerative process .t /, t  0, with the regeneration times k , k D 0; 1; : : : . We also assume the following condition to hold, which excludes the degenerate case where the regeneration times take zero value with probability 1: R3 : fL D Pf 1 > 0g > 0.

3.1 Regenerative processes and regenerative stopping times

107

By conditions R1 and R2 , the random variables n  n1 , n  1, are independent and identically distributed. Thus, conditions R1 –R3 imply that P

n ! 1 as n ! 1:

(3.1.1)

Let now  be a random variable defined on the same probability space, .; F ; P /, and taking values in the interval Œ0; 1. We say that  is a regenerative stopping time if it regenerates together with the process .t /, t  0, at times n , n D 0; 1; : : :, in the sense that .n/

R4 : The conditional probabilities Pf .n/ .tr / 2 Ar ; r  sr ; r D 1; : : : ; m;  

n C t =B;   n g D Pf .tr / 2 Br ; r  sr ; r D 1; : : : ; m;   t g, Ar 2 , tr ; sr  0, r D 1; : : : ; m, m  1, t  0, are the same for any B 2 Fn./ and for every n D 0; 1; : : : . We also assume the following condition that excludes the degenerate case where  < 1 with probability 1, R5 : f D Pf < 1 g 2 Œ0; 1/. In what follows we refer to f as a stopping probability in one regeneration period. Let .t /, t  0, be a regenerative process with regeneration times n , n D 0; 1; : : :, and  be a regenerative stopping moment that satisfies conditions R1 –R5 . Then, for any A 2 , the probabilities P .t; A/ D Pf .t / 2 A;  > t g and q.t; A/ D Pf .t / 2 A; 1 > t;  > t g are measurable functions of t and P .t; A/ satisfies the following renewal equation: Z t P .t  s; A/F .ds/; t  0; (3.1.2) P .t; A/ D q.t; A/ C 0

where F .t / D Pf 1  t;   1 g:

(3.1.3)

Note that F .t / is possibly an improper distribution function such that (a) F .0/ D Pf 1 D 0g D 1  fL < 1 and (b) F .1/ D Pf  1 g D 1  f > 0. Relation (3.1.2) can be used to simplify the definition of a regenerative process and a regenerative stopping time. One can say that  is a regenerative stopping time for a regenerative process .t /; t  0, with regeneration times n , n D 0; 1; : : :, if relation (3.1.2) holds.

3.1.2 Regenerative processes with transition period A regenerative process with transition period is a useful generalisation of a regenerative process.

108

3 Nonlinearly perturbed regenerative processes

The definition differs of the one above only in the formulation of condition R2 . This condition requires that the corresponding joint finite dimensional distributions would have an identity for every n D 0; 1; : : : . A version of this condition adapted to the case of a regenerative process with a transition period does require an identity for the corresponding joint finite dimensional distributions for every n D 1; 2; : : :: .n/

R02 : The joint finite dimensional distributions Pf .n/ .tr / 2 Ar ; r  sr ; r D .1/ 1; : : : ; mg D Pf .1/ .tr / 2 Ar ; r  sr ; r D 1; : : : ; mg, Ar 2 , tr ; sr  0, r D 1; : : : ; m, m  1 are the same for every n D 1; 2; : : : . According to condition R02 , the initial process .t /; t  0, can possess finite dimensional distributions which differ from the corresponding finite dimensional distributions of the shifted processes . n C t /, t  0, for n D 1; 2; : : : . Thus one can refer to the period Œ0; 1 / as a transition period. Condition R3 should be replaced with the following condition: .1/ R03 : fL D Pf 1 D 2  1 > 0g > 0.

Similar changes should be made in the definition of regenerative stopping moments. The definition differ from the one above only in the formulation of condition R4 . This condition requires an identity for the corresponding conditional probabilities for every n D 0; 1; : : : . A version of this condition adapted to the case of a regenerative process and regenerative stopping times with a transition period does require an identity for the corresponding conditional probabilities only for every n D 1; 2; : : :: .n/

R04 : The conditional probabilities Pf .n/ .tr / 2 Ar ; r  sr ; r D 1; : : : ; m;   .1/

n C t =B;   n g D Pf .1/ .tr / 2 Br ; r  sr ; r D 1; : : : ; m;   1 C t =B;   1 g, Ar 2 , tr ; sr  0, r D 1; : : : ; m, m  1, t  0, are the same for ./ any B 2 F n and for every n D 1; 2; : : : . Condition R5 should be replaced with the following condition that excludes the degenerate case (where the regenerative stopping time  < 1 with probability 1): R05 : fQ D Pf < 1 g 2 Œ0; 1/. This condition should also be supplemented with the following condition as to exclude the degenerate case (where the regenerative stopping time  < 2 with probability 1): R005 : f D Pf  1 < 2 =  1 g 2 Œ0; 1/. Let .t /, t  0, be a regenerative process with a transition period and regeneration times n , n D 0; 1; : : :, and  be a regenerative stopping time that satisfies conditions R1 and R02 –R05 , as well as R005 . Introduce the probabilities PQ .t; A/ D Pf .t / 2 A;  > t g and q.t; Q A/ D Pf .t / 2 A; 1 > t;  > t g, and let P .t; A/ D

109

3.1 Regenerative processes and regenerative stopping times .1/

Pf .1/ .t / 2 A;   1 > t =  1 g and q.t; A/ D Pf .1/ .t / 2 A; 1 > t;   1 > t =  1 g. Then, for any A 2 , these probabilities are measurable functions of t , and PQ .t; A/ and P .t; A/ satisfy the following renewal relation: Z t PQ .t; A/ D q.t; Q A/ C P .t  s; A/FQ .ds/; t  0; (3.1.4) 0

where FQ .t / D Pf 1  t;   1 g;

(3.1.5)

together with the following renewal equation: Z t P .t  s; A/F .ds/; P .t; A/ D q.t; A/ C 0

t  0;

(3.1.6)

where F .t / D Pf 1.1/  t;   1  1.1/ =  1 g:

(3.1.7)

Relations (3.1.4) and (3.1.6) can be used to simplify the definition of a regenerative process and a regenerative stopping time with transition period. One can say that  is a regenerative stopping time for a regenerative process .t /, t  0, with regeneration times n , n D 0; 1; : : :, if relations (3.1.4) and (3.1.6) hold.

3.1.3 The structure of regenerative stopping times The following lemma follows from the definition of regenerative stopping times. Lemma 3.1.1. Let conditions R1 –R5 hold. Then Pf  n g D .1  f /n ;

n D 0; 1; : : : :

(3.1.8)

k

Proof. Denote x fhg D hŒx= h. Obviously, . nC1  n /f2 g , k  1, is a sequence of a.s. k monotonically non-decreasing random variables, and . nC1  n /f2 g ! . nC1 

n / as k ! 1. Using these facts and condition R4 we get, for n D 0; 1; : : :, that Pf  nC1 =  n g D Pf  n C . nC1  n /=  n g k g

D lim Pf  n C . nC1  n /f2 k!1

D lim

k!1

D lim

k!1

1 X

Pf  n C r2k ; r2k  nC1  n < .r C 1/2k =  n g

r D0 1 X

Pf  r2k ; r2k  1 < .r C 1/2k g

r D0 f2k g

D lim Pf  1 k!1

=  n g

g D Pf  1 g D 1  f:

(3.1.9)

110

3 Nonlinearly perturbed regenerative processes

Using relation (3.1.9) we get, for every n D 0; 1; : : :, that Pf  n g D

n1 Y

Pf  kC1 =  k g D .1  f /n :

(3.1.10)

kD0

The proof is complete. Remark 3.1.1. In the case of a regenerative process with a transition period, formula (3.1.8) takes the following form: Pf  n g D .1  fQ/.1  f /n1 ;

n D 1; 2; : : : :

(3.1.11)

The following lemma is a direct corollary of Lemma 3.1.1. Lemma 3.1.2. Let conditions R1 –R5 hold. Then one of the two alternatives holds: (a) Pf D 1g D 1 if f D 0, or (b) Pf < 1g D 1 if f > 0. The only nontrivial case is where f > 0. Let us consider it in more details. We introduce the random variables  D min.n  1 W n1   < n /; D   1 ; n D n  n1 ;

n D 1; 2; : : : :

It is obvious that the regenerative stopping time  can be represented in the form of the random sum D

1 X

n C :

(3.1.12)

nD1

It follows from Lemma 3.1.9 that the random variable  has a geometric distribution with the parameter f , i.e., Pf > ng D .1  f /n ;

n D 0; 1; : : : :

(3.1.13)

Also, the random variables n D n  n1 , n D 1; 2; : : :, are i.i.d. random variables. Thus  is a geometric sum of i.i.d. random variables. The representation (3.1.12) has, however, a drawback. The random variables , , and n  n1 , n D 1; 2; : : :, employed in this representation, are not independent. Fortunately, this defect can be overcomed. Let us consider the random variable  ^ 1 . The distribution function of this random variable can be represented in the form of a linear combination of two conditional distribution functions, FO .t / D Pf ^ 1  t g D FO .t /.1  f / C FOC .t /f;

t  0;

(3.1.14)

111

3.1 Regenerative processes and regenerative stopping times

where FO .t / D Pf 1  t =  1 g D F .t /=.1  f /; FOC .t / D Pf  t = < 1 g D .F .t /  FO .t //=f: In the case where the stopping probability is f D 0, the distribution function FOC .t / can be taken arbitrarily. This does not affect formula (3.1.14). For simplicity, we take FOC .t / D Œ0;1/ .t / in this case. Lemma 3.1.3. Let conditions R1 –R5 hold, and f > 0. Then, for every 0  t; t1 ; : : : ; tn < 1, n D 1; 2; : : :, Pf D n;  t; k  k1  tk ; k D 1; : : : ; n  1g D .FO .t /  F .t //

n1 Y

F .tk /

kD1 n1

D f .1  f /

FOC .t /

n1 Y

FO .tk /:

(3.1.15)

kD1

Proof. By conditions of the lemma, 0 < f < 1. Using the definition of the random variables , , and n  n1 , n D 1; 2; : : :, and condition R4 we get that Pf D n;  t; k  k1  tk ; k D 1; : : : ; n  1g D Pf n1   < n ;   n1  t; k  k1  tk ; k D 1; : : : ; n  1g D Pf  n1  t;  < n = n1  ; k  k1  tk ; k D 1; : : : ; n  1g  Pf n1  ; k  k1  tk ; k D 1; : : : ; n  1g D Pf  t;  < 1 gPf n1  ; k  k1  tk ; k D 1; : : : ; n  1g D .FO .t /  F .t //Pf n1  n2  tn1 ;

n1  = n2  ; k  k1  tk ; k D 1; : : : ; n  2g  Pf n2  ; k  k1  tk ; k D 1; : : : ; n  2g D .FO .t /  F .t //Pf 1  tn1 ; 1  g  Pf n2  ; k  k1  tk ; k D 1; : : : ; n  2g D .FO .t /  F .t //F .tn1 /Pf n2  ; k  k1  tk ; k D 1; : : : ; n  2g D ::: D .FO .t /  F .t //

n1 Y kD1

This proves the lemma.

n1

F .tk / D f .1  f /

FOC .t /

n1 Y kD1

FO .tk /:

(3.1.16)

112

3 Nonlinearly perturbed regenerative processes

Lemma 3.1.3 permits to represent the regenerative stopping time  in the following d form (the symbol D means that the random variables on the left and the right-hand sides have the same distribution): d

 D

1 O X

O n C ; O

(3.1.17)

nD1

where (a) O is a geometrically distributed random variable with the parameter f , (b) the random variable O has the distribution function FO C .t /, (c) the random variables

O n , n D 1; 2; : : :, have the distribution function FO  .t /, (d) the random variables ; O O and O n , n D 1; 2; : : :, are mutually independent.

3.2 Mixed ergodic and limit theorems for perturbed regenerative processes In this section we present mixed ergodic and limit theorems for perturbed regenerative processes which describe pseudo-stationary phenomena for nonlinearly perturbed regenerative processes.

3.2.1 Mixed ergodic and limit theorems for perturbed regenerative processes Let for every "  0, ."/ .t /, t  0, be a regenerative process on a measurable phase space X (with the corresponding -algebra  of measurable subsets) with regeneration times 0 D 0."/  1."/  : : :, and let ."/ be a random variable defined on the same probability space and taking values in the interval Œ0; 1. We assume that the random variable ."/ is a regenerative stopping time for the regenerative process ."/ .t /, t  0, for every "  0, and thus conditions R1 –R5 hold for every "  0. In this case, according to the definition given above, for any A 2 , the probabilities ."/ ."/ P .t; A/ D Pf ."/ .t / 2 A; ."/ > t g and q ."/ .t; A/ D Pf ."/ .t / 2 A; 1 > t; ."/ > t g are measurable functions of t , and P ."/ .t; A/ satisfies the renewal equation Z t P ."/ .t  s; A/F ."/ .ds/; t  0; (3.2.1) P ."/ .t; A/ D q ."/ .t; A/ C 0

where ."/

."/

F ."/ .t / D Pf 1  t; ."/  1 g:

(3.2.2)

An important quantity is the stopping probability in one regeneration period f ."/ , which coincides with the deficiency of the distribution F ."/ .t /, ."/

f ."/ D Pf."/ < 1 g D 1  F ."/ .1/:

3.2

113

Mixed ergodic and limit theorems for perturbed regenerative processes

We also use the notations, m."/ r

D

."/ E. 1 /r .."/



."/

1 /

Z D

1

0

s r F ."/ .ds/;

r  1:

To the renewal equation (3.2.1) we apply the results obtained in Chapters 1 and 2, which will yield various variants of asymptotics for the probabilities P ."/ .t; A/. We impose the following condition: D13 : F ."/ ./ ) F .0/ ./ as " ! 0, where F .0/ .s/ is a proper non-arithmetic distribution function. Denote

."/ ."/ FL ."/ .t / D Pf 1  t g; FO ."/ .t / D Pf."/ ^ 1  t g:

By definition, for any t  0, ."/

."/

1  f ."/  F ."/ .t / D Pf 1 > t; ."/  1 g ."/  1  FO ."/ .t / D Pf."/ ^ 1 > t g ."/

."/

."/

."/

D Pf 1 > t; ."/  1 g C Pf1 > t; ."/ < 1 g  1  FL ."/ .t / D Pf 1."/ > t g:

(3.2.3)

Note also that .1  FL ."/ .t //  .1  f ."/  F ."/ .t // D Pf 1."/ > t; ."/ < 1."/ g  Pf."/ < 1."/ g D f ."/ :

(3.2.4)

Condition D13 implies that f ."/ ! f .0/ D 0 as " ! 0:

(3.2.5)

Also, condition D13 and relations (3.2.3)–(3.2.5) imply that FO ."/ ./ ) FO .0/ ./  F .0/ ./ as " ! 0

(3.2.6)

FL ."/ ./ ) FL .0/ ./  F .0/ ./ as " ! 0:

(3.2.7)

and

We also assume the following condition to hold: R1 M8 : limT !1 lim0"!0 T .1  FO ."/ .s// ds D 0. Condition M8 , due to estimates (3.2.3), implies the following condition: R1 M9 : limT !1 lim0"!0 T .1  f ."/  F ."/ .s// ds < 1;

114

3 Nonlinearly perturbed regenerative processes

and itself is a consequence of the following condition: R1 M10 : limT !1 lim0"!0 T .1  FL ."/ .s// ds < 1. Note also that conditions D11 and M9 imply that ."/

.0/

m1 ! m1 < 1 as " ! 0:

(3.2.8)

We also assume the following condition imposed on the forcing function in the renewal equation (3.1.2): F7 : There exists a class of sets 0  such that, for every A 2 0 , the following relation holds: limu!0 lim0"!0 supjvju jq ."/ .t C v; A/  q .0/ .t; A/j D 0 almost everywhere with respect to the Lebesgue measure on Œ0; 1/. Let us also assume the following condition that balances the rate of growth of time t ."/ ! 1 and the rate of the perturbation determined, in this case, by the speed with which the stopping probability f ."/ vanish, B4 : 0  t ."/ ! 1 and f ."/ ! 0 as " ! 0 in such a way that f ."/ t ."/ ! , where 0    C1. Note that condition B4 implies the following: (a) if  > 0, then automatically we have f ."/ > 0 for all " small enough, (b) if there exists a sequence 0 < "n ! 0 as n ! 1 such that f ."n / D 0; n  1, then  D 0. Denote R 1 ."/ q .s; A/m.ds/ ."/  .A/ D 0 ; A 2 : (3.2.9) ."/ m1 Note that  ."/ .A/ is a probability distribution, i.e.,  ."/ .X/ D 1 if and only if f D 0. In particular, f .0/ D 0 by condition D13 . In this case,  .0/ .A/ is a stationary distribution of the regenerative process .0/ .t /, t  0. In the next subsection, we will give a more detailed comment. Let us formulate a theorem which is similar to Theorem 1.3.1 applied to the renewal equation (3.2.1). ."/

Theorem 3.2.1. Let conditions D13 , M8 , F7 hold, and 0  t ."/ ! 1 as " ! 0 such that condition B4 holds. Then, for any A 2 0 , P ."/ .t ."/ ; A/ D Pf ."/ .t ."/ / 2 A; ."/ > t ."/ g .0/

!  .0/ .A/e =m1

as " ! 0:

(3.2.10)

3.2

Mixed ergodic and limit theorems for perturbed regenerative processes

115

Proof. The theorem follows directly from Theorem 1.3.1. Conditions D13 and B4 coincide, respectively, with conditions D6 and B1 for the distribution functions F ."/ .t / defined above. Also, conditions M8 and F7 imply that condition F2 holds for the forcing functions q ."/ .t; A/. Condition F7 coincides in this case with condition F2 .a/. Obviously, for any A 2  and t  0, ."/

."/

q ."/ .t; A/ D Pf ."/ .t / 2 A; ."/ ^ 1 > t g  Pf."/ ^ 1 > t g D 1  FO ."/ .t /:

(3.2.11)

It follows from (3.2.11) that condition M8 implies that conditions F2 .b/ and F2 (c) hold. The first implication is obvious and the second one follows from the asymptotic estimates X lim lim h sup jq ."/ .t; A/j T !1 0"!0

r T = h r ht.rC1/h

Z

 lim

lim

1

T !1 0"!0 T h

.1  FO ."/ .s// ds D 0:

(3.2.12)

So, all the conditions of Theorem 1.3.1 hold. By applying this theorem to the renewal equation (3.1.2) we complete the proof.

3.2.2 Ergodic theorems for perturbed regeneration processes Let us consider a model for which the regenerative stopping moment ."/ D 1 with probability 1 for all " small enough, say "  "1 . As follows from Lemma 3.1.2, this is equivalent to the following condition: D14 : f ."/ D 0 for every "  "1 . In this case, condition B4 automatically holds for any 0  t ."/ ! 1 as " ! 0 and  D 0. Theorem 3.2.1 yields in this case a variant of the ergodic theorem for perturbed regenerative processes. Theorem 3.2.2. Let conditions D13 , D14 , M8 , and F7 hold. Then, for any 0  t ."/ ! 1 as " ! 0, P ."/ .t ."/ ; A/ D Pf ."/ .t ."/ / 2 Ag !  .0/ .A/ as " ! 0; A 2 0 :

(3.2.13)

It is worth noting that, in the case where the regenerative processes ."/ .t /  ."/ .0/ .0/ .t /; t  0, and the regenerative times k  k , k D 0; 1; : : :, for all "  0, i.e., they do not depend on the parameter ", the conditions of Theorem 3.2.2 reduce to the

116

3 Nonlinearly perturbed regenerative processes

standard conditions of the ergodic theorem for the non-perturbed regenerative process .0/ .t /; t  0, which is governed by the classical renewal theorem. These conditions are: (a) the distribution function FL .0/ .t / D Pf 1.0/  t g is non- arithmetic; (b) FL .0/ .t / has a finite first moment; (c) the function q .0/ .t; A/ D Pf .0/ .t / 2 A; 1.0/ > t g is continuous almost everywhere with respect to the Lebesgue measure on Œ0; 1/ for A 2 0 .

3.2.3 Stationary distributions for perturbed regenerative processes We continue to consider a model such that condition D14 holds. Let us also assume the following conditions: D15 : F ."/ .s/ is a non-arithmetic distribution function for all " small enough, say "  "2 ; and F8 : q ."/ .s; A/ is directly Riemann integrable on Œ0; 1/ for A 2 0 and all " small enough, say "  "3 . Note that, under condition M8 , it is enough to assume that for A 2 0 the function q ."/ .s; A/ is continuous almost everywhere with respect to the Lebesgue measure on Œ0; 1/ for all " small enough. In this case, condition F8 holds. The Lebesgue integration can be replaced with the Riemann integration in the right-hand side of (3.2.9). Condition M8 implies that m."/ 1 < 1 for all " small enough, say "  "4 . Let us define, "0 D min."1 ; "2 ; "3 ; "4 /:

(3.2.14)

The renewal Theorem 1.1.1 implies the following statement. Theorem 3.2.3. Let conditions D14 , D15 , M8 , and F8 hold. Then, for "  "0 , Pf ."/ .t / 2 Ag !  ."/ .A/ as t ! 1;

A 2 0 :

(3.2.15)

The expression for  ."/ .A/ given by formula (3.2.9) defines a probability distribution, which, due to asymptotic relation (3.2.15), is called a stationary distribution for the regenerative process ."/ .t /, t  0. It should be noted that, under condition M8 , the expression for  ."/ .A/ given by formula (3.2.9) defines a probability distribution for "  "0 even if condition F10 does not hold. This is so, since we use in this formula the Lebesgue integration. Moreover it can be proved that the average value of the Pf ."/ .s/ 2 Ag over the interval Œ0; t  converges to  ."/ .A/ as t ! 1.

3.2

Mixed ergodic and limit theorems for perturbed regenerative processes

117

3.2.4 Convergence of stationary distributions for perturbed regenerative processes The following theorem supplements the asymptotic relations (3.2.13) and (3.2.15). Theorem 3.2.4. Let conditions D13 , D14 , M8 , and F7 hold. Then  ."/ .A/ !  .0/ .A/ as " ! 0;

A 2 0 :

(3.2.16)

Proof. The theorem follows from Theorem 1.2.4, since conditions D13 , D14 , M8 , and F7 imply that conditions D2 , M2 , and F2 hold for the distribution functions F ."/ .t / and the forcing functions q ."/ .t; A/. Note that convergence in (3.2.16) is guaranteed for A 2 0 , while the stationary distribution is defined for A 2 . This is true, because condition F7 provides convergence of the corresponding forcing functions only for A 2 0 . However, in most applications, 0 D .

3.2.5 Weak convergence of distributions for regenerative stopping times Let us return to a model for which condition D14 is not assumed to hold. Moreover, we assume a condition that is opposite in some sense: D16 : f ."/ > 0 for every " > 0. Note that f .0/ D 0, according to condition D13 . We consider the case where A D X. Obviously, P ."/ .t; X/ D Pf."/ > t g and q ."/ .t; X/ D Pf 1."/ > t; ."/ > t g D 1  FO ."/ .t /. Condition D13 implies in this case that condition F7 holds. Indeed, relation (3.2.6) implies that 1  FO ."/ .t / ! 1  F .0/ .t / as " ! 0 for every point of continuity of the limit function. Moreover, since the functions 1  FO ."/ .t / are monotone, this convergence is locally uniform in every such point. But, the set S0 of continuity points of the function 1  F .0/ .t / is the interval Œ0; 1/ except for at most a countable set. Thus the Lebesgue measure is m.S 0 / D 0. Note also that condition B4 holds for t ."/  t =f ."/ as " ! 0 for every t > 0. In this case,  D t . Finally,  .0/ .X/ D 1, as was mentioned in Subsection 3.2.1. Theorem 3.2.1 and the remarks above yield in this case that, under conditions D13 , M8 , and D16 , .0/

P ."/ .t =f ."/ ; X/ D Pff ."/ ."/ > t g ! e t=m1

as " ! 0; t  0:

(3.2.17)

Relation (3.2.17) shows that the regenerative stopping moment f ."/ ."/ , normalised by the quantity f ."/ , has an exponential limit distribution with the parameter .0/ 1=m1 .

118

3 Nonlinearly perturbed regenerative processes

Marginal asymptotic relations (3.2.13) and (3.2.17) give a clear interpretation to the mixed asymptotic relation (3.2.10) given in Theorem 3.2.1. This theorem is a mixed ergodic and limit theorem for perturbed regenerative processes and regenerative stopping times. The asymptotic relation (3.2.10) shows that the state of the perturbed regenerative process ."/ .t =f ."/ / at the moment t =f ."/ and the normalised regenerative stopping time f ."/ ."/ are asymptotically independent and the joint limit distribution is the product of the stationary distribution  .0/ .A/ for the limit regenerative process .0/ and the limit exponential distribution e t=m1 for the normalised regenerative stopping times. As far as the marginal asymptotic relation (3.2.17) is concerned, it can be obtained under conditions weaker than the conditions D13 and M8 used in Theorem 3.2.1. To improve these conditions, one can use representation (3.1.17) for the regenerative stopping times " in the form of the random sum and to apply general theorems for randomly stopped stochastic processes, for example, given in Silvestrov (1974, 2004a). Let us briefly describe this procedure. ."/

Let O ."/ , O ."/ and O k , k  1, be mutually independent random variables which are used to construct representation (3.1.17) for the regenerative processes ."/ .t /, t  0, and the regenerative stopping times ."/ . Let us first consider the case where conditions D16 and M8 remain true while condition D13 is weaken by omitting in it the requirement that the limit distribution should be non-arithmetic. Condition D13 , reduced as described above, implies, due to relations (3.2.5) and (3.2.6), that (a) F."/ ./ ) F .0/ ./ as " ! 0, where F."/ .t / D Pf 1."/  t =."/  ."/

1 g. This relation, estimate (3.2.3), and condition M8 also imply that (b) m."/  D R 1 ."/ R 1 .0/ .0/ .ds/ as " ! 0. Relations (a) and (b) are suf0 sF .ds/ ! m1 D 0 sF ficient for fulfilment of the weak law of large numbers for the sums of i.i.d. random variables O k."/ , k  1, with the distribution function F."/ .t /. Considered in a stochastic processes setting, this limit theorem takes the form of the relation (c) P d ."/ .0/

."/ .t / D ktn."/ O k =n."/ ; t  0 ! .0/ .t / D t m1 ; t  0, that holds for any non-random functions 0 < n."/ ! 1 as " ! 0. We are interested in the case where n."/ D .f ."/ /1 and will take this function in what follows. The random variable O ."/ has a geometric distribution with the parameter f ."/ . ."/ Thus, Pff ."/ O ."/ > t g D .1  f ."/ /Œt=f  ! e t as " ! 0 for t  0. In other d

words, this means that (e)  ."/ D f ."/ O ."/ !  .0/ as " ! 0, where  .0/ is a random variable exponentially distributed with the parameter 1. The process .0/ .t / is a non-random function, relations (c) and (e), due to the wellknown Slutsky theorem (see, for example, Silvestrov (2004a), page 12), imply also d

that (f) ." ; ."/ .t //, t  0 ! . .0/ ; .0/ .t //, t  0, as " ! 0.

3.2

Mixed ergodic and limit theorems for perturbed regenerative processes

119

Since the processes ."/ .t /, t  0, are non-decreasing and the limit process .0/ .t / is also continuous, relation (f) implies (see, Silvestrov (2004a), Theorem 2.2.3 and the d

remarks in Subsection 2.2.5) that (g) ."/ . ."/ / ! .0/ . .0/ / as " ! 0. It remains to note that condition M8 implies, due to estimate (3.2.3), that 1  R1 ."/ O FC .t =f ."/ /  .1  FO ."/ .t =f ."/ //=f ."/  t 1 t=f ."/ .1  FO ."/ .s// ds ! 0 as " ! 0, t > 0, where FO ."/ .t / D Pf."/  t =."/ < ."/ g and FO ."/ .t / D Pf."/ ^ ."/  t g. C

In other words, this means that (h)

."/

Df

1 ."/ O ."/

1

P

! 0 as " ! 0. d

Relations (g) and (h) imply, again due to Slutsky theorem, that ."/ . ."/ /C ."/ ! d

.0/ . .0/ / as " ! 0. According to (3.1.17), we have f ."/ ."/ D ."/ . ."/ / C ."/ . d

Thus, (i) f ."/ ."/ ! .0/ . .0/ / as " ! 0. It remains to note that the random variable .0/ .0/

.0/ . .0/ / D m1  .0/ has an exponential distribution with the parameter 1=m1 . The proof presented above suggests a way in which one can achieve further improvements in the conditions that provide asymptotic relation (3.2.17). First of all, conditions D13 and M8 can be replaced with the well-known conditions in the main criterion for convergence, which would provide necessary and sufficient conditions for the weak law of large number to hold for sums of i.i.d. random variables in a triangular array mode. Recall that we consider processes of ."/.t / D P ."/ O k =n."/ , t  0 in the case where n."/ D 1=f ."/ . These conditions (see, ktn."/ for example, Lo`eve (1955)) are: (j) .1  F."/ .t =f ."/ /=f ."/ ! 0 as " ! 0, t > 0, R 1=f ."/ ."/ .0/ sF .ds/ ! m.0/ and (k) 0 1 2 .0; 1/ as " ! 0. Note that, in this case, m1 may be not the first moment of the distribution function F .0/ .t / but just some positive constant. Conditions (j) and (k), implied by conditions D13 (the reduced version) and M8 , are necessary and sufficient for relation (c) to hold. Thus, by repeating the proof described above, we get relation (g). Condition M8 is also used for getting relation (h). As follows from the proof of this relation, it is also implied by the weaker relation ."/ .t =f ."/ / ! 0 as " ! 0. (l) 1  FOC Summarizing the remarks above we can conclude that conditions (j)–(l) imply relation (i) that is equivalent to the asymptotic relation (3.2.17). As was pointed out is Subsection 3.2.4, conditions D13 and M8 used in the mixed ergodic and limit Theorem 3.2.1 serve very well in the marginal ergodic Theorem 3.2.2. In particular, they reduce to the best known ergodic conditions used in the classical renewal theorem for a non-perturbed model. At the same time, the remarks above show that these conditions can be weakened in the marginal limit theorem for regenerative stopping times. The question arises of why this is so? As a matter of fact, these conditions, combined with the asymptotic relation (3.2.10), should serve in mixed ergodic/limit theorems for both the ergodic and the limit theorems. The ergodic theorems do require conditions stronger than those for the limit theorems. This explains why these con-

120

3 Nonlinearly perturbed regenerative processes

ditions can be improved when applying to marginal limit theorems for regenerative stopping times. We shall see in the next section that the situation is different in the case where the transition phenomena are studied under Cram´er type conditions. In this case, we shall get asymptotic relations that are suitable for mixed ergodic theorems for perturbed regenerative processes with large deviation and for theorems with regenerative stopping times. The latter theorems can not be improved in the way described above. In this case, both types of marginal ergodic and large deviation theorems do require the same type of conditions as the corresponding mixed ergodic and large deviation theorems.

3.2.6 Mixed ergodic and limit theorems for perturbed regenerative processes with transition period In this model, Theorem 3.2.1 can be directly applied to the shifted regenerative processes, which means that the probabilities P ."/ .t; A/ and q ."/ .t; A/, and the distribution functions F ."/ .t /, FL ."/ .t / and FO ."/ .t / introduced in Subsection 3.2.1, should be replaced in the renewal equation (3.2.1), conditions D13 , M8 , F7 , B4 , and Theorem 3.2.1 with the probabilities P ."/ .t; A/ D Pf ."/ . 1."/ C t / 2 A, ."/  1."/ > t =."/ 

1."/ g, q ."/ .t; A/ D Pf ."/ . 1."/ C t / 2 A, 2."/  1."/ > t , ."/  1."/ > t =."/  1."/ g, ."/ ."/ ."/ ."/ and the distribution functions F ."/ .t / D Pf 2  1  t , ."/  1  2  ."/ ."/ ."/ ."/ ."/

1 =."/  1 g, FL ."/ .t / D Pf 2  1  t g, and FO ."/ .t / D Pf.."/  1 / ^ ."/ ."/ ."/ . 2  1 /  t =."/  1 g. Theorem 3.2.1, implies in this case that, under conditions D13 , M8 , and F7 , and 0  t ."/ ! 1 satisfying condition B4 , for any A 2 0 , we have .0/

P ."/ .t ."/ ; A/ !  .0/ .A/e =m1 where

R1 

."/

.A/ D

0

q ."/ .s; A/m.ds/ ."/ m1

."/

as " ! 0;

; m1 D

Z 0

1

(3.2.18)

sF ."/ .ds/:

As far as non-shifted regenerative processes are concerned, the probabilities P ."/ .t; A/ and q ."/ .t; A/, the distribution functions F ."/ .t /, FL ."/ .t / and FO ."/ .t /, introduced in Subsection 3.2.1, should be replaced with the probabilities PQ ."/ .t; A/ D ."/ Pf ."/ .t / 2 A; ."/ > t g and qQ ."/ .t; A/ D Pf ."/ .t / 2 A; 1 > t; ."/ > t g and the distribution functions FQ ."/ .t / D Pf 1."/  t , ."/  1."/ g, FLQ ."/ .t / D Pf 1."/  t g, and ."/ ."/ FOQ ."/ .t / D Pf."/ ^ 1  t g. Let us also denote fQ."/ D Pf."/ < 1 g D FQ ."/ .1/. The probabilities PQ ."/ .t; A/ are connected with the probabilities P ."/ .t; A/ by the renewal type relation Z t ."/ ."/ Q P .t; A/ D qQ .t; A/ C P ."/ .t  s; A/FQ ."/ .ds/; t  0: (3.2.19) 0

3.2

121

Mixed ergodic and limit theorems for perturbed regenerative processes

It is clear that some additional appropriate conditions should be imposed on the functions qQ ."/ .t; A/ and the distribution functions FQ ."/ .t / and FOQ ."/ .t / in order to get, for the probabilities PQ ."/ .t; A/, asymptotic relation analogous to (3.2.18). These conditions are: D17 : fQ."/ ! fQ.0/ 2 Œ0; 1 as " ! 0; and

D18 : limT !1 lim"!0 .1  FOQ ."/ .T / D 0. The following estimate is analogous to (3.2.3), 1  fQ."/  FQ ."/ .t / D Pf 1."/ > t; ."/  1."/ g ."/  1  FOQ ."/ .t / D Pf."/ ^ 1 > t g ."/

."/

."/

."/

D Pf 1 > t; ."/  1 g C Pf1 > t; ."/ < 1 g  1  FLQ ."/ .t / D Pf 1."/ > t g:

(3.2.20)

This estimate implies that condition D18 is a consequence of the following condition: D19 : limT !1 lim"!0 .1  FLQ ."/ .T / D 0. The estimate (3.2.20) also implies that the following condition can be used, together with condition D18 , instead of condition D17 : D20 : FQ ."/ ./ ) FQ .0/ ./ as " ! 0, where F .0/ .s/ is a proper or improper distribution function. Note that condition D20 does require convergence of the pre-limit distribution functions FQ ."/ .s/ to FQ .0/ .s/ as " ! 0 in continuity points s of the limit distribution function, but does not require that the values FQ ."/ .1/ D 1  fQ."/ would converge to FQ .0/ .1/ D 1  fQ.0/ as " ! 0. Lemma 3.2.1. Let condition D18 hold. Then, condition D20 implies condition D17 . Proof. Let Tn , n  1, be a sequence of continuity points of the distribution function FQ ."/ .t / such that Tn ! 1 as n ! 1. Using estimate (3.2.20) and conditions D18 and D20, we get lim jfQ."/  fQ.0/ j "!0

 lim j1  fQ."/  FQ ."/ .Tn /j C jFQ ."/ .Tn /  FQ .0/ .Tn /j "!0

C j1  fQ.0/  FQ .0/ .Tn /j

 lim .1  FOQ ."/ .Tn // C j1  fQ.0/  FQ .0/ .Tn /j ! 0 as n ! 1: "!0

This relation proves the lemma.

(3.2.21)

122

3 Nonlinearly perturbed regenerative processes

It is useful to note that the distribution FQ .0/ .s/ in condition D20 is proper if and only if the limit constant is fQ.0/ D 0 in condition D17 . Note also that we allow for the degenerate case where fQ.0/ D 1, i.e., where condition R05 does holds for the limit regenerative process. Theorem 3.2.1 takes the following form. Theorem 3.2.5. Let conditions D13 , M8 , F7 , D17 , D18 hold, and 0  t ."/ ! 1 as " ! 0 such that condition B4 holds. Then, for any A 2 0 , Q ."/ ; A/ D Pf ."/ .t ."/ / 2 A; ."/ > t ."/ g P.t .0/

! .1  fQ.0/ / .0/ .A/e =m1

as " ! 0:

(3.2.22)

Proof. The proof is based on the use of the asymptotic relation (3.2.10) given in Theorem 3.2.1 and the renewal relation (3.2.19). Note that relation (3.2.10) is implied by conditions D13 , M8 , F7 , and by the assumption that 0  t ."/ ! 1 as " ! 0 such that condition B4 holds. Condition D18 implies that, for any 0  t ."/ ! 1 as " ! 0 and any A 2 0 , Q ."/ ; A/  P.t

Z

t ."/ 0

P ."/ .t ."/  s; A/FQ ."/ .ds/

D qQ ."/ .t ."/ ; A/  1  FOQ ."/ .t ."/ / ! 0 as " ! 0:

(3.2.23)

Relation (3.2.23) implies that to get (3.2.22) we should prove that, for any 0  t ."/ ! 1 as " ! 0 such that condition B4 holds and any A 2 0 , Z

t ."/

0

.0/

P ."/ .t ."/  s; A/FQ ."/ .ds/ ! .1  fQ.0/ / .0/ .A/e =m1 as " ! 0: (3.2.24)

Two cases should be considered: (a) fQ.0/ D 1 or (b) fQ.0/ D 1. The case (a) is trivial. Indeed in this case we get, for any A 2 0 , Z

t ."/ 0

P ."/ .t ."/  s; A/FQ ."/ .ds/  1  fQ."/ ! 0 as " ! 0:

(3.2.25)

Let us consider the case (b). Let us use Lemma 1.2.2 for asymptotic analysis of the integral in the right-hand side of (3.2.24). Note first that this integral can be rewritten in the form where the limits of integration do not depend on ", Z

t ."/ 0

P

."/

.t

."/

 s; A/FQ ."/ .ds/ D

Z

1 0

P ."/ .t ."/  s; A/FQ ."/ .ds/;

where one should take P ."/ .t  s; A/ D 0 for s > t .

(3.2.26)

3.2

Mixed ergodic and limit theorems for perturbed regenerative processes

123

For every A 2 0 , the functions P ."/ .t ."/  s; A/ are asymptotically bounded, i.e., lim

sup

"!0 0st ."/

P ."/ .t ."/  s; A/  1:

(3.2.27)

Let us show that, for every A 2 0 and any s  0 and v > 0, .0/

sup jP ."/ .t ."/  s C u; A/   .0/ .A/e =m1 j ! 0 as " ! 0:

(3.2.28)

jujv

Suppose that asymptotic relation (3.2.28) does not take place. This means that there exists a sequence 0 < "k ! 0 as k ! 1 and uk with juk j  v such that .0/

jP ."k / .t ."k /  s C uk ; A/   .0/ .A/e =m1 j ¹ 0 as " ! 0:

(3.2.29)

Since juk j  v and f ."k / ! 0 as k ! 1, condition B4 implies that f ."/ .t ."k /  s C uk / !  as k ! 1 for any s  0 and v > 0. Therefore, by (3.2.19), .0/

jP ."k / .t ."k /  s C uk ; A/   .0/ .A/e =m1 j ! 0 as " ! 0:

(3.2.30)

Relation (3.2.29) contradicts (3.2.30). Thus, relation (3.2.28) holds. Let us now assume that condition D20 holds, instead of condition D17 . Conditions D18 and D20 imply, by Lemma 3.2.1, that 1  fQ."/ D FQ ."/ .1/ ! 1  fQ.0/ D FQ .0/ .1/ as " ! 0:

(3.2.31)

Condition D20 and relations (3.2.27), (3.2.28) and (3.2.31) imply that for every A 2 0 , the functions P ."/ .t ."/  s; A/ and the measures FQ ."/ .ds/ satisfy conditions of Lemma 1.2.2. Thus, we get by applying Lemma 1.2.2 that, for any A 2 0 , Z 1 Z 1 .0/ P ."/ .t ."/  s; A/FQ ."/ .ds/ !  .0/ .A/e =m1 FQ .0/ .ds/ 0

0

.0/

D .1  fQ.0/ / .0/ .A/e =m1

as " ! 0: (3.2.32)

Let us consider now the case where condition D17 holds. We show that the proof can be reduced to the case considered above, i.e., to the case where condition D20 holds. Indeed, let us take an arbitrary sequence 0 < "n ! 0 as n ! 1. It is always possible to select from this sequence a subsequence "nk ! 0 as k ! 1 such that FQ ."nk / ./ ) FQ ./ as " ! 0; where FQ .s/ is a proper or improper distribution function.

(3.2.33)

124

3 Nonlinearly perturbed regenerative processes

The proof given above for the case where condition D20 holds can be applied to a subsequence of models with the perturbation parameters "nk , k  1, which and to yields, for any A 2 0 , the following relation: Z 1 P ."nk / .t ."nk /  s; A/FQ ."nk / .ds/ 0

.0/

! .1  fQ/ .0/ .A/e =m1

as " ! 0;

(3.2.34)

where fQ D 1  FQ .1/. The limit distribution FQ .s/ does depend on the choice of the sequence "n and the subsequence "nk , but the limit fQ does not. Indeed, conditions D18 and relation (3.2.33) imply, by Lemma 3.2.1, that 1  FQ ."nk / .1/ D fQ."nk / ! 1  FQ .1/ D fQ as " ! 0:

(3.2.35)

On the other hand, condition D17 implies that 1  FQ ."nk / .1/ D fQ."nk / ! fQ.0/ as " ! 0:

(3.2.36)

Thus, fQ D fQ.0/ and, therefore, the limit in the right hand-side of (3.2.34) does not depend on the choice of the sequence "n and the subsequence "nk . Moreover, it coincides with the limit in the right-hand side of relation (3.2.24). This implies that relation (3.2.24) holds. The proof of the theorem is complete.

3.2.7 Ergodic theorems for perturbed regenerative processes with transition period Let us again consider a model in which the regenerative stopping time is ."/ D 1 with probability 1 for every "  0. In this case, f ."/  0, F ."/ .s/ D FO ."/ .s/ D D E. 2."/  1."/ /. Also, q ."/ .s; A/ D FL ."/ .s/ D Pf 2."/  1."/  sg and m."/ 1 ."/ ."/ ."/ Pf ."/ . 1 C t / 2 A; 2  1 > t g. Condition B4 automatically holds for any ."/ ! 1 as " ! 0 and  D 0. As far as characteristics of the transition 0  t period are concerned, in this case, FQ ."/ .s/ D FOQ ."/ .s/ D FLQ ."/ .s/ D Pf 1."/  sg, qQ ."/ .s; A/ D Pf ."/ .t / 2 A; 1."/ > t g and fQ."/ D 0 for every "  0. Note that, in this case, condition D17 automatically holds with the limit constant fQ.0/ D 0, conditions D18 and D19 are equivalent, and condition D20 implies condition D18 . Theorem 3.2.5 yields in this case a variant of the ergodic theorem for perturbed regenerative processes. Theorem 3.2.6. Let conditions D13 , D14 , M8 , F7 , D19 hold. Then, for any 0  t ."/ ! 1 as " ! 0, P ."/ .t ."/ ; A/ D Pf ."/ .t ."/ / 2 Ag !  .0/ .A/ as " ! 0; A 2 0 :

(3.2.37)

3.3 Mixed ergodic and large deviation theorems for regenerative processes

125

Conditions D17 and D18 hold automatically for any fixed "  0. Thus, convergence Pf ."/ .t / 2 Ag to a stationary distribution  ."/ .A/ is also preserved for regenerative processes with transition period. Let "0 be the parameter defined in Theorem 3.2.3. Theorem 3.2.7. Let conditions D14 , D15 , M8 , and F8 hold. Then, for any all "  "0 , Pf ."/ .t / 2 Ag !  ."/ .A/ as t ! 1;

A 2 0 :

(3.2.38)

3.3 Mixed ergodic and large deviation theorems for perturbed regenerative processes In this section we give mixed ergodic and large deviation theorems for perturbed regenerative processes and regenerative stopping times. These theorems describe pseudostationary and quasi-stationary phenomena for perturbed regenerative processes.

3.3.1 Pseudo-stationary exponential asymptotics for perturbed regenerative processes Let us now assume the condition: C3 : There exists ı > 0 such that lim0"!0

R1 0

e ıs FO ."/ .ds/ < 1.

Obviously, Ee

."/

ı1

.

."/



."/

1 /

Z DE

1 0

e ıs F ."/ .ds/  Ee ı.

."/

."/ ^ ."/ / 1

."/

."/

."/

D Ee ı1 .."/  1 / C Ee ı1 .."/ < 1 / Z 1 Z 1 ."/ ıs O ."/ ı1 e F .ds/  Ee D e ıs FL ."/ .ds/: (3.3.1) D 0

0

Therefore, condition C3 implies the following condition: R1 C4 : There exists ı > 0 such that lim0"!0 0 e ıs F ."/ .ds/ < 1. At the same time, condition C3 is implied by the following condition: R1 C5 : There exists ı > 0 such that lim0"!0 0 e ıs FL ."/ .ds/ < 1. Z

Denote 

."/

./ D

1 0

e s F ."/ .ds/:

The following equation plays an important role in what follows:  ."/ ./ D 1:

(3.3.2)

126

3 Nonlinearly perturbed regenerative processes

As follows from Lemma 1.4.1, conditions D13 and C3 imply that there exists a unique nonnegative root ."/ of equation (3.3.2) for all " small enough, and ."/ ! 0 as " ! 0. ."/ .0/ Note also that conditions D13 and C3 imply that m1 ! m1 < 1 as " ! 0. Let us formulate a theorem which is an analogue of Theorem 1.4.1 applied to renewal equation (3.2.1). Theorem 3.3.1. Let conditions D13 , C3 , and F7 hold. Then, for any 0  t ."/ ! 1 as " ! 0 and any A 2 0 , Pf ."/ .t ."/ / 2 A; ."/ > t ."/ g !  .0/ .A/ as " ! 0: expf."/ t ."/ g

(3.3.3)

Proof. Condition D13 implies that condition D6 holds for the distribution functions F ."/ .s/. Also, condition C3 implies condition C4 , which in this case coincides with condition C1 . Condition F7 implies that, for every A 2 0 , condition F5 .a/ holds for the function ."/ q .s; A/. Since q ."/ .s; A/  1, condition F5 .b/ obviously holds. Let us show now that condition C3 implies that for every A 2 0 condition F5 .c/ holds for the functions q ."/ .s; A/ for any 0 <  < ı. Indeed, it obviously follows from condition C3 that for any " small enough, Z 1 ıT ."/ O e .1  F .T //  e ıs FO ."/ .ds/ ! 0 as T ! 1: (3.3.4) T

Using this relation, integration by parts and condition C3 we have X lim lim h sup e t jq ."/ .t; A/j T !1 0"!0

r T = h r ht.rC1/h

Z

 lim

lim

1

T !1 0"!0 T h

 lim

lim e

e s .1  FO ."/ .s//ds

.ı /.T h/

T !1 0"!0

 lim e .ı /.T h/ lim

Z

T h Z 1

0"!0 0

T !1

1

e ıs .1  FO ."/ .s//ds

e ıs .1  FO ."/ .s//ds

 D lim e .ı /.T h/ lim ı 1 .1  FO ."/ .0// 0"!0 T !1 Z 1  e ıs FO ."/ .ds/ C ı 1 0

 lim e .ı /.T h/ .ı 1 C ı 1 lim T !1

Z

0"!0 0

1

e ıs FO ."/ .ds// D 0:

Now we can finish the proof of the theorem by applying Theorem 1.4.1.

(3.3.5)

3.3 Mixed ergodic and large deviation theorems for regenerative processes

127

3.3.2 Quasi-stationary exponential asymptotics for perturbed regenerative processes Let us now consider the case where the limit distribution F .0/ .t / can be either proper or improper. We use the representation ."/ ."/ FO."/ .t / D Pf 1  t =."/  1 g D F ."/ .t /=.1  f ."/ /;

t  0:

The following condition replaces D13 : D21 :

(a) FO."/ ./ ) FO.0/ ./ as " ! 0, where FO.0/ .t / is a non-arithmetic distribution function; (b) f ."/ ! f .0/ 2 Œ0; 1/ as " ! 0.

We assume also the following version of the Cram´er type condition: C6 : There exists ı > 0 such that: R1 (a) lim0"!0 0 e ıs FO ."/ .ds/ < 1; R1 (b) 0 e ıs F .0/ .ds/ > 1. Let us consider the condition: D22 : F ."/ ./ ) F .0/ ./ as " ! 0, where F .0/ .s/ is a proper or improper distribution not concentrated in zero, i.e., such that F .0/ .0/ < F .0/ .1/, and FO.0/ .t / D F .0/ .t /=F .0/ .1/ is a non-arithmetic distribution function. As follows from Lemma 1.4.4, conditions D22 and C4 , which are weaker than C6 , imply that f ."/ ! f .0/ 2 Œ0; 1/ as " ! 0. Therefore, under condition C6 , condition D21 is implied by the simpler condition D22 . Let us again consider the characteristic equation (3.3.2). It follows from Lemma 1.4.3 that under conditions D21 and C6 there exists a unique nonnegative root ."/ of equation (3.3.2) for all " small enough. Note that ."/ D 0 if and only if f ."/ D 0 or, equivalently, ."/ > 0 if and only if f ."/ > 0. Also, ."/ ! .0/ as " ! 0. Note also that .0/ < ı. It is also useful to note that, in the case f .0/ D 0, conditions D21 and C6 .a/ imply condition C6 .b/. Denote R 1 ."/ s ."/ e q .s; A/m.ds/ ."/ Q .A/ D 0 R 1 ; A 2 : (3.3.6) ."/ s  F ."/ .ds/ 0 se Let us formulate a theorem which is analogous to Theorem 1.4.2 applied to the renewal equation (3.2.1). This theorem covers both cases, f .0/ > 0 and f .0/ D 0.

128

3 Nonlinearly perturbed regenerative processes

Recall that f .0/ > 0 if and only if .0/ > 0 or, equivalently, f .0/ D 0 if and only if .0/ D 0. Theorem formulated below generalises Theorem 3.3.1 and reduces to this theorem in the case where f .0/ D 0. Theorem 3.3.2. Let conditions D21 , C6 , and F7 hold. Then, for any 0  t ."/ ! 1 as " ! 0 and any A 2 0 , Pf ."/ .t ."/ / 2 A; ."/ > t ."/ g ! Q .0/ .A/ as " ! 0: expf."/ t ."/ g

(3.3.7)

Proof. Condition D21 implies that condition D11 holds for the distribution functions F ."/ .s/. Also, condition C6 implies condition C4 that, in this case, coincides with condition C2 . Condition F7 implies that for every A 2 0 , condition F6 .a/ holds for the function q ."/ .s; A/. Since 0  q ."/ .s; A/  1, condition F6 .b/ obviously holds. Let us show now that condition C6 implies that, for every A 2 0 , condition F6 .c/ holds for the functions q ."/ .s; A/ if 0 < .0/ C  < ı. Indeed, we have similarly to (3.3.5) that X .0/ sup e . C /t jq ."/ .t; A/j lim lim h T !1 0"!0

r T = h r ht.rC1/h

Z

1

.0/ C /s

.1  FO ."/ .s//ds Z 1 .ı..0/ C //.T h/  lim e lim e ıs .1  FO ."/ .s//ds 0"!0 0 T !1 Z 1 .0/  lim e .ı. C //.T h/ .ı 1 C ı 1 lim e ıs FO ."/ .ds// D 0: (3.3.8)

 lim

lim

T !1 0"!0 T h

T !1

e .

0"!0 0

Application of Theorem 1.4.2 finishes the proof.

3.3.3 Quasi-stationary distributions for perturbed regenerative processes Condition D21 does not imply that the distribution function FO."/ .t / is non-arithmetic for all " small enough. Let us assume the following condition: D23 : FO."/ .t / is a non-arithmetic distribution function for all " small enough, say "  "1 . Let us also assume the following condition: F9 : There exists  > 0 such that .0/ C  < ı and, for every A 2 0 , the func.0/ tion e . C /s q ."/ .s; A/ is directly Riemann integrable on Œ0; 1/ for all " small enough, say "  "2 .

3.3 Mixed ergodic and large deviation theorems for regenerative processes

129

Note that under condition C6 it is sufficient to assume that, for A 2 0 , the function q ."/ .s; A/ is continuous almost everywhere with respect to Lebesgue measure on Œ0; 1/ for all " small enough. In this case, condition F9 holds and Lebesgue integration can be replaced with RiemannRintegration in the right-hand side of (3.3.6). 1 Condition C6 implies that 0 e ıs FO ."/ .ds/ 2 .1; 1/ for all " small enough, say "  "3 . Let "0 D min."1 ; "2 ; "3 /:

(3.3.9)

The following theorem is a direct corollary of Theorem 3.3.2. Theorem 3.3.3. Let conditions D23 , C6 , and F9 hold. Then, for "  "0 , e

."/ t

Pf ."/ .t / 2 A; ."/ > t g ! Q ."/ .A/ as t ! 1;

A 2 0 :

(3.3.10)

Note that, under conditions D21 and C6 , the expression for Q ."/ .A/ given by formula (3.3.6) is finite for all "  0 even if condition F9 does not hold. This is so, since this formula uses Lebesgue integration. It should be noted that, in general, Q ."/ .A/ may not be a probability distribution, i.e., it may be such that 0 < Q ."/ .X/ ¤ 1. Let us now define 

."/

R 1 ."/ s ."/ e q .s; A/m.ds/ Q ."/ .A/ ; .A/ D ."/ D R01 ."/  s .1  FO ."/ .s//ds Q .X/ 0 e

A 2 :

(3.3.11)

Note that our notations are consistent with those introduced in formula (3.2.9). Indeed, in the case considered in (3.2.9), ."/ D 0 and 1  FO ."/ .s/  1  F ."/ .s/. In ."/ this case, the denominator in (3.3.11) is equal to m1 . By the definition,  ."/ .A/ is a probability distribution. It is called a quasi-stationary distribution for the regenerative process ."/ .t /, t  0, with the regenerative stopping time ."/ . Theorem 3.3.4. Let conditions D21 , C6 , and F9 hold. Then, for "  "0 , Pf ."/ .t / 2 A=."/ > t g D

e

."/ t

Pf ."/ .t / 2 A; ."/ > t g e ."/ t Pf."/ > t g

!  ."/ .A/ as t ! 1;

A 2 0 :

(3.3.12)

Relation (3.3.12) clarifies why it is natural to call the distribution  ."/ .A/ a quasistationary distribution for the regenerative process ."/ .t / with the regenerative stopping time ."/ .

130

3 Nonlinearly perturbed regenerative processes

3.3.4 Convergence of quasi-stationary distributions for perturbed regenerative processes The following theorem supplements the asymptotic relations (3.3.7) and (3.3.10). Theorem 3.3.5. Let conditions D21 , C6 , and F7 hold. Then Q ."/ .A/ ! Q .0/ .A/ as " ! 0;

A 2 0 :

(3.3.13)

Proof. As was shown in the proof of Theorem 3.3.2, conditions D21 , C6 and F7 imply that conditions D11 , C2 and F6 of Theorem 1.4.1 hold for the distribution functions F ."/ .s/ and the functions q ."/ .s; A/ for every A 2 0 . So we can prove the theorem by applying Theorem 1.4.4. The following theorem is a direct corollary of formula (3.3.11) and Theorem 3.3.5. Theorem 3.3.6. Let conditions D21 , C6 , and F7 hold. Then  ."/ .A/ !  .0/ .A/ as " ! 0;

A 2 0 :

(3.3.14)

Remark 3.3.1. It is useful to note that formula (3.3.6) that defines the quantities Q ."/ .A/ and formula (3.3.11) that defines the quasi-stationary probabilities  ."/ .A/ do not require for the distribution F ."/ .t / to be non-arithmetic. This remark allows to drop the requirement that the limit distribution F .0/ .t / be non-arithmetic in condition D21 used in Theorems 3.3.5 and 3.3.6.

3.3.5 Large deviation asymptotics for distributions of regenerative stopping moments We assume that condition D16 holds and consider the case where A D X. As was pointed out in Subsection 3.2.4, P ."/ .t; X/ D Pf."/ > t g and q ."/ .t; X/ D Pf."/ ^ ."/ ."/ ."/

1 > t g D 1  FO ."/ .t /. Recall also that F ."/ .t / D Pf 1  t; 1  ."/ g. Condition F7 is equivalent in this case to the following condition: D24 : FO ."/ ./ ) FO .0/ ./ as " ! 0, where FO .0/ .t / is a proper distribution function. Indeed, condition F7 obviously implies that FO ."/ .t / ! FO .0/ .t / as " ! 0 almost everywhere with respect to Lebesgue measure on Œ0; 1/. Let S00 be the corresponding set of convergence. Obviously, S00 is dense in the interval Œ0; 1/. Therefore, condition F10 holds, since weak convergence of distribution functions concentrated on the interval Œ0; 1/ is implied by their pointwise convergence in points of any set dense in this interval. On the other hand, condition D24 implies that 1  FO ."/ .t / ! 1  FO .0/ .t / as " ! 0 for every point of continuity of the limit function. Moreover, since the functions 1  FO ."/ .t / are monotone, this convergence is locally uniform for every such point.

3.3 Mixed ergodic and large deviation theorems for regenerative processes

131

But the set S0 of continuity points of the function 1  F .0/ .t / is the interval Œ0; 1/ except for at most a countable set. Thus the Lebesgue measure m.S 0 / D 0. Therefore, condition F7 holds. Note also that, in the case where f .0/ D 0, condition D21 implies, due to relation (3.2.6), that condition D24 holds. Also, in this case, FO.0/ .t /  FO .0/ .t /  F .0/ .t /. Define R 1 ."/ s e .1  FO ."/ .s//ds ."/ : (3.3.15) Q .X/ D 0 R 1 ."/ s F ."/ .ds/ 0 se Formula (3.3.15) is a particular case of formula (3.3.6) taken for A D X. Note that Lebesgue integration, used in the integral in the numerator of (3.3.6), is replaced with ."/ Riemann integration in (3.3.15). As a matter of fact, the function e  s .1  FO ."/ .s// has at most countable set of discontinuity points. As was mentioned above, it may happen that Q ."/ .X/ ¤ 1. However, if f ."/ D 0, then ."/ D 0 and FO ."/ .t /  F ."/ .t /. In this case, R1 .1  F ."/ .s//ds ."/ Q .X/ D 0 R 1 ."/ D 1: .ds/ 0 sF Theorem 3.3.2 takes in this case the following form. Theorem 3.3.7. Let conditions D21 , C6 , and D24 hold. Then for any 0  t ."/ ! 1 as " ! 0, Pf."/ > t ."/ g ! Q .0/ .X/ as " ! 0: expf."/ t ."/ g

(3.3.16)

Asymptotic relation (3.3.16) becomes trivial if condition D14 holds, i.e., f ."/ D 0 for " small enough, say "  "0 . Indeed, in this case, ."/ D 1 with probability 1 for every "  "0 . In this case, both expressions in the left and the right hand-sides of (3.3.16) equal to 1. Let us consider the case where condition D16 holds, i.e., f ."/ > 0 for " > 0. There are two alternative cases: (i) f .0/ D 0 and (ii) f .0/ > 0. Let us first consider case (i). Condition D21 implies that f ."/ ! 0 as " ! 0. Therefore, ."/ ! 0 as " ! 0. Also, in this case, Q .0/ .X/ D 1. Choose t ."/ ! 1 such that ."/ t ."/ ! 0 as " ! 0. Now, the asymptotic relation (3.3.16) takes the following form: Pf."/ > t ."/ g ! 1 as " ! 0:

(3.3.17)

Relation (3.3.17) implies that, in case (i), the stopping times ."/ are asymptotically stochastically unbounded random variables as " ! 0, i.e., P

."/ ! 1 as " ! 0:

(3.3.18)

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3 Nonlinearly perturbed regenerative processes

Let us take t ."/ D t =."/ ! 1 as " ! 0, where t > 0. Asymptotic relation (3.3.16) takes the following form: Pf."/ ."/ > t g ! 1 as " ! 0; expft g

t > 0:

(3.3.19)

This is a weak convergence asymptotic relation. Indeed, relation (3.3.19) means that d

the random variables converge, ."/ ."/ ! 0 as " ! 0, where 0 is an exponentially distributed random variable with parameter 1. Note that (3.3.19) is consistent with the asymptotic relation (3.2.17). According ."/ .0/ to Lemma 1.4.2, in this case, ."/  f ."/ =m."/ 1 as " ! 0. Also, m1 ! m1 as " ! 0. Thus, (3.3.19) can be rewritten in the equivalent form Pff ."/ ."/ > t g ! .0/ expft =m1 g as " ! 0, t > 0, identical to (3.2.17). It should be noted however that asymptotic relation (3.2.17) was obtained under weaker conditions than (3.3.19). We can, however, take t ."/ D tQ."/ =."/ , where tQ."/ ! 1 as " ! 0. In this case (3.3.16) takes the form of a large deviation asymptotic relation, Pf."/ ."/ > tQ."/ g ! 1 as " ! 0: expftQ."/ g

(3.3.20)

Asymptotic relation (3.3.20) is much stronger than (3.3.19). This relation yields that the asymptotic relative error for approximation of the tail probabilities P f."/ ."/ > tQ."/ g with the corresponding tail probabilities of the limit exponential distribution equals 0 for any tQ."/ ! 1. This is achieved because of the choice of the very wellfitted normalising coefficients ."/ . The payment for this result is a need to use the Cram´er type condition C6 . The drawback of (3.3.20) is that the normalising coefficient ."/ is given as a root of the nonlinear equation (3.3.2). The question arises whether it is possible to ."/ replace ."/ in (3.3.20), for example, with a simpler coefficient f ."/ =m1 and to ."/ ."/ rewrite this relation in the form Pff ."/ ."/ > tQ."/ m1 g= expftQ."/ m1 g ! 1 as ."/ " ! 0? The answer is partly affirmative. Relation ."/  f ."/ =m1 means that ."/ /. Thus, (3.3.20) implies that Pff ."/ ."/ > tQ."/ m."/ g  ."/ D f ."/ =m."/ 1 C o.f 1 ."/ g  expf.f ."/ =m."/ C o.f ."/ //tQ."/ m."/ =f ."/ g. Therefore, expf."/ tQ."/ m."/ =f 1 1 1 the suggested replacement would additionally require to assume that tQ."/ o.f ."/ /=f ."/ ! 0, i.e., to restrict the rate of convergence of tQ."/ to 1. The questions related to the use of normalising coefficients given in a more explicit form will be investigated in the next section where we shall get asymptotic expansions for the roots ."/ in asymptotic power series. Let us now consider case (ii). Condition D21 implies in this case that f ."/ ! .0/ f > 0 as " ! 0. Therefore, ."/ ! .0/ > 0 as " ! 0. Also, in this case, it is possible that Q .0/ .X/ ¤ 1.

133

3.3 Mixed ergodic and large deviation theorems for regenerative processes

Let us also write the representation formula (3.1.14) for the regeneration processes ."/ .t /, t  0, and the regenerative stopping times ."/ , ."/ FO ."/ .t / D FO."/ .t /.1  f ."/ / C FOC .t /f ."/ ;

where

."/ ."/ FOC .t / D Pf."/  t =."/ < 1 g;

t  0;

(3.3.21)

t  0:

Representation (3.3.21) implies that, under conditions D21 and D24, ."/ .0/ FOC ./ ) FOC ./ as " ! 0:

(3.3.22)

."/

Let also O ."/ ; O ."/ and O k , k  1, be mutually independent random variables that are used to construct representation (3.1.17) for the regenerative processes ."/ .t /, t  0, and the regenerative stopping times ."/ . R1 ."/ ."/ .s/ D E expfs O 1."/ g D 0 e st FO."/ .dt / and C .s/ D Denote by  R 1 st ."/ ."/ O E expfs O g D e F .dt / the Laplace transforms for, respectively, the ran."/

O 1

0

C

."/

dom variables and O . In terms of the Laplace transforms, representation (3.1.17) takes, for the regenerative processes ."/ .t /, t  0, and the regenerative stopping times ."/ , the following form: ."/

E expfs."/ g D

C .s/f ."/ ."/ .s/ 1  .1  f ."/ /

;

s  0: ."/

(3.3.23) .0/

Conditions D21 , D24 , and relation (3.3.22) imply that C .s/ ! C .s/ and ."/ .s/ !  .0/ .s/ as " ! 0; s  0. Also, as was mentioned above, f ."/ ! f .0/ > 0   as " ! 0. Thus, it follows from (3.3.23) that E expfs."/ g ! E expfs.0/ g as " ! 0, s  0. Therefore, d

."/ ! .0/ as " ! 0:

(3.3.24)

Relation (3.3.24) implies that, in case (ii), the stopping times ."/ are asymptotically stochastically bounded random variables as " ! 0, i.e., lim lim Pf."/ > T g D 0:

T !1 "!0

(3.3.25)

Under conditions D21 , C6 , and D24 , relation (3.3.16) yields, in case (ii), that for any 0  t ."/ ! 1 as " ! 0, Pf."/ > t ."/ g ! Q .0/ .X/ as " ! 0: expf."/ t ."/ g

(3.3.26)

134

3 Nonlinearly perturbed regenerative processes

Note that in case (ii), ."/ ! .0/ > 0, thus the tail probabilities for stopping times have exponential decay with parameter ."/ , asymptotically, as " ! 0, separated from zero. In conclusion, let us see what form will the corresponding conditions take in the ."/ .0/ case where the processes ."/ .t /  .0/ .t /, t  0, the regeneration times k  k , k D 0; 1; : : :, and the regenerative stopping times ."/  .0/ for all "  0, i.e., if they do not depend on the parameter ". Condition D24 reduces to the assumptions that (a) F.0/ .t / is non-arithmetic, and (b) f .0/ > 0; condition D24 automatically holds; R 1 and condition C6 reduces to the assumption that (c) there exists ı > 0 such that 0 e ıs F .0/ .ds/ 2 .1; 1/. Relation (3.3.16) takes the following form: Pf.0/ > t g ! 1 as t ! 1: Q .0/ .X/ expf.0/ t g

(3.3.27)

Asymptotic relation (3.3.27) means that the tail probabilities P f.0/ > t g have exponential decay and can be approximated by the corresponding tail probabilities for an .0/ exponential type distribution with the corresponding tail probabilities Q .0/ .X/e  t . The asymptotic relative error of such an approximation, as t ! 1, equals 0. The remarks made in Subsections 3.3.3 and 3.3.4 permit to interpret the mixed asymptotic relation (3.3.7) given in Theorem 3.3.2 as a mixed ergodic and large deviation theorem for perturbed regenerative processes and regenerative stopping times.

3.3.6 Pseudo-stationary exponential asymptotics for perturbed regenerative processes with transition period Let us consider the model of perturbed regenerative processes with transition period introduced in Subsection 3.2.5. We use all the notations introduced in this section. We assume the following two conditions: R1 C7 : There exists 0 < 1 < ı such that lim0"!0 0 e 1 s FQ ."/ .ds/ < 1; and D25 : There exists 0 < 2 < ı such that limT !1 lim"!0 e 2 T .1  FOQ ."/ .T // D 0. Note that estimate (3.3.1) implies that both conditions C7 and D25 can be replaced with the following condition: R 1 s O ."/ C8 : There exists 0 <  < ı such that lim0"!0 e FQ .ds/ < 1. 0

More precisely, C7 and D25 hold for any 0 < 1 ; 2 <  . Let us formulate a theorem that generalises Theorem 3.3.1 to the case of perturbed regeneration processes with transition period and asymptotically proper distribution functions F ."/ .t /.

3.3 Mixed ergodic and large deviation theorems for regenerative processes

135

Theorem 3.3.8. Let conditions D13 , C3 , F7 , D17 , C7 , and D25 hold. Then, for any 0  t ."/ ! 1 as " ! 0 and any A 2 0 , Pf ."/ .t ."/ / 2 A; ."/ > t ."/ g ! .1  f .0/ / .0/ .A/ as " ! 0: expf."/ t ."/ g

(3.3.28)

Proof. The proof is based on the use of the renewal relation (3.2.19) and the asymptotic relation (3.3.3) given in Theorem 3.3.1. Note that relation (1.4.9) is provided by conditions D13 , C3 , F7 , and that 0  t ."/ ! 1 as " ! 0 in an arbitrary way in this relation. Renewal relation (3.2.19) can be rewritten in the following form: e

."/ t

Q ."/ ; A/ D e ."/ t qQ ."/ .t ."/ ; A/ P.t Z t ."/ ."/ ."/ e  .ts/ P ."/ .t ."/  s; A/e  s FQ ."/ .ds/; t  0: (3.3.29) C 0

Conditions D13 and C3 imply that ."/ ! 0 as " ! 0. Therefore, ."/  1 ^ 2 for all " small enough. Thus we get by D25 that, for any 0  t ."/ ! 1 as " ! 0 and any A 2 0 , e

."/ t ."/

Q ."/ ; A/  P.t De

Z

t ."/

0

."/ t ."/ ."/

qQ

e

."/ .t ."/ s/

P ."/ .t ."/  s; A/e 

."/ s

FQ ."/ .ds/

."/ .t ."/ ; A/  e 2 t .1  FOQ ."/ .t ."/ // ! 0 as " ! 0: (3.3.30)

Relation (3.3.30) implies that, in order to get (3.3.28), we should prove that, for any 0  t ."/ ! 1 as " ! 0 and any A 2 0 , Z

t ."/ 0

e

."/ .t ."/ s/

P ."/ .t ."/  s; A/e 

."/ s

FQ ."/ .ds/

! .1  f .0/ / .0/ .A/ as " ! 0: Let us also show that, for every A 2 0 , the functions e  are asymptotically bounded, i.e., lim

sup

"!0 0st ."/

e

."/ .t ."/ s/

."/ .t ."/ s/

(3.3.31) P ."/ .t ."/  s; A/

P ."/ .t ."/  s; A/ D R < 1:

(3.3.32)

Let us suppose, conversely, that there exists 0  s ."/  t ."/ such that e

."/ .t ."/ s ."/ /

P ."/ .t ."/  s ."/ ; A/ ! 1 as " ! 0:

(3.3.33)

We can suppose that t ."/  s ."/ ! w as " ! 0, where 0  w  1 (if not we can use subsequences).

136

3 Nonlinearly perturbed regenerative processes

The limit constant w can not be finite, since in this case, ."/ .t ."/ s ."/ / ! 0w D 0 as " ! 0 and, therefore, in contradiction with (3.3.33), lim e 

."/ .t ."/ s ."/ /

"!0

P ."/ .t ."/  s ."/ ; A/  lim e 

."/ .t ."/ s ."/ /

"!0

D 1:

(3.3.34)

Neither can w be equal to 1. Indeed, in this case, u."/ D t ."/  s ."/ ! 1 as " ! 0. Then it would follow from (3.3.3) that e

."/ u."/

P ."/ .u."/ ; A/ !  .0/ .A/ as " ! 0;

(3.3.35)

which would again contradict (3.3.33). Therefore, relation (3.3.33) can not hold but, on the contrary, relation (3.3.32) holds. Two following cases should be considered: (a) fQ.0/ D 1 and (b) fQ.0/ D 1. Let us consider case (a). Using that ."/  1 ^ 2 for all " small enough, relation (3.3.32), and inequality FQ ."/ .T /  FQ ."/ .1/ D 1  fQ."/ , we get that Z 1 ."/ ."/ ."/ lim e  .t s/ P ."/ .t ."/  s; A/e  s FQ ."/ .ds/ "!0 0

Z  lim R "!0

1

0

 lim R.e

1 T

"!0

e

."/ s

FQ ."/ .T / C e .ı 1 /T

D lim Re .ı 1 /T "!0

FQ ."/ .ds/

Z

1 0

Z

1 T

e ıs FQ ."/ .ds//

e ıs FQ ."/ .ds/ ! 0 as T ! 1:

(3.3.36)

Consider case (b). Let us apply Lemma 1.2.2 for an asymptotic analysis of the integral in the right-hand side of (3.3.30). To do this, note first that the integral in (3.3.31) can be rewritten in the form where the limits of integration do not depend on ", Z t ."/ ."/ ."/ ."/ e  .t s/ P ."/ .t ."/  s; A/e  s FQ ."/ .ds/ 0

Z D

1 0

e

."/ .t ."/ s/

P ."/ .t ."/  s; A/e 

."/ s

FQ ."/ .ds/;

(3.3.37)

where one should take P ."/ .t  s; A/ D 0 for s > t . Let us show that, for every A 2 0 and any s  0 and v > 0, sup je 

."/ .t ."/ sCu/

P ."/ .t ."/  s C u; A/   .0/ .A/j ! 0 as " ! 0:

(3.3.38)

jujv

Suppose that asymptotic relation (3.2.26) does not take place. This means that there exists a sequence 0 < "k ! 0 as k ! 1 and uk with juk j  v such that je 

."/ .t ."/ sCu/

P ."k / .t ."k /  s C uk ; A/   .0/ .A/j ¹ 0 as " ! 0:

(3.3.39)

137

3.3 Mixed ergodic and large deviation theorems for regenerative processes

Since the sequence juk j  v is bounded, .t ."k /  s C uk / ! 1 as k ! 1 for any s  0 and v > 0. Therefore, by (3.3.3), je 

."/ .t ."/ sCu/

P ."k / .t ."k /  s C uk ; A/   .0/ .A/j ! 0 as " ! 0:

(3.3.40)

Relation (3.3.39) contradicts (3.3.40). Thus, relation (3.3.38) holds. Introduce the distribution functions Z t ."/ e  s FQ ."/ .ds/; t  0: FQQ ."/ .t / D 0

Condition C7 and the relation ."/ ! 0 as " ! 0 imply that there exists 0 < 10 < 1 such that ."/ C 10  1 for all " small enough, and thus, Z 1 0 lim e 1 s FQQ ."/ .ds/ 0"!0 0

Z

 lim

0"!0 0

1

e 1 s FQ ."/ .ds/ < 1:

(3.3.41)

Relation (3.3.41) implies that FQQ ."/ .ds/ is a finite measure for " small enough. Let us now assume that condition D20 holds instead of condition D17 . Condition D20 and the relation ."/ ! 0 as " ! 0 imply that FQQ ."/ ./ ) FQ .0/ ./ as " ! 0:

(3.3.42)

Relations (3.3.41) and (3.3.42) imply, by Lemma 1.4.4, that FQQ ."/ .1/ ! FQ .0/ .1/ as " ! 0:

(3.3.43)

Condition D20 and relations (3.3.32), (3.3.38), and (3.3.42) imply that, for any 0  t ."/ ! 1 as " ! 0 and any A 2 0 , conditions of Lemma 1.2.2 hold for the functions ."/ ."/ e  .t s/ P ."/ .t ."/  s; A/ and the measures FQQ ."/ .ds/. Thus we get, by applying Lemma 1.2.2, that for any 0  t ."/ ! 1 as " ! 0 and any A 2 0 , Z 1 ."/ ."/ ."/ e  .t s/ P ."/ .t ."/  s; A/e  s FQ ."/ .ds/ 0 Z 1 ."/ ."/ D e  .t s/ P ."/ .t ."/  s; A/FQQ ."/ .ds/ 0 Z 1 !  .0/ .A/FQ .0/ .ds/ D .1  f .0/ / .0/ .A/ as " ! 0: (3.3.44) 0

Let us consider now the case where condition D17 holds. We show that the proof can be reduced to the case considered above, i.e., to the case where condition D20 holds.

138

3 Nonlinearly perturbed regenerative processes

Indeed, let us take an arbitrary sequence 0 < "n ! 0 as n ! 1. It is always possible to choose from this sequence a subsequence "nk ! 0 as k ! 1 such that FQ ."nk / ./ ) FQ ./ as k ! 1;

(3.3.45)

where FQ .s/ is a proper or improper distribution function. Relation (3.3.45) and the relation ."/ ! 0 as " ! 0 imply that FQQ ."nk / ./ ) FQ ./ as k ! 1:

(3.3.46)

The proof given above for the case where condition D20 holds can be applied to a subsequence of models with the perturbation parameters "nk , k  1, to yield, for any A 2 0 , the following relation: Z

1 0

P ."nk / .t ."nk /  s; A/FQQ ."nk / .ds/ ! .1  fQ/ .0/ .A/ as k ! 1;

(3.3.47)

where fQ D 1  FQ .1/. The limit distribution FQ .s/ does depend on the choice of the sequence "n and the subsequence "nk , but the limit fQ does not. Indeed, relations (3.3.41) and (3.3.46), by Lemma 1.4.4, imply that 1  FQQ ."nk / .1/ D fQQ."nk / ! 1  FQ .1/ D fQ as k ! 1:

(3.3.48)

On the other hand, by condition D17 , 1  FQQ ."nk / .1/ D fQQ."nk / ! fQ.0/ as k ! 1:

(3.3.49)

Thus, fQ D fQ.0/ and, therefore, the limit in the right-hand side of (3.3.47) does not depend on the choice of the sequence "n and the subsequence "nk . Moreover, it coincides with the limit in the right-hand side of relation (3.3.44). This implies that relation (3.3.44) holds. The proof of the theorem is complete.

3.3.7 Quasi-stationary exponential asymptotics for perturbed regenerative processes with transition period Let us continue to consider the model of perturbed regenerative processes with transition period, introduced in Subsection 3.2.5. We use all the notations introduced there. However, we consider now the case where FQ ."/ .t / are asymptotically proper or improper distribution functions.

139

3.3 Mixed ergodic and large deviation theorems for regenerative processes

In this case we impose, on the characteristics of the transition period, the following Cram´er type condition: C9 : There exists 0 <  0 < ı  .0/ such that lim0"!0 1;

R1 0

e .

.0/ C 0 /s

FQ ."/ .ds/
t ."/ g ! Q .0/ Q .0/ .A/ as " ! 0: expf."/ t ."/ g

(3.3.52)

Proof. The proof is similar to the proof of Theorem 3.3.8 and is based on the use of the renewal relation (3.2.19) and the asymptotic relation (3.3.3) given in Theorem 3.3.1. Note that relation (3.3.3) is implied by conditions D21 , C6 , F7 , and that 0  t ."/ ! 1 as " ! 0 in an arbitrary way in this relation. Let us again rewrite the renewal relation (3.2.19) in the following form: e

."/ t

Q ."/ ; A/ D e ."/ t qQ ."/ .t ."/ ; A/ P.t Z t ."/ ."/ ."/ e  .ts/ P ."/ .t ."/  s; A/e  s FQ ."/ .ds/; t  0: (3.3.53) C 0

Conditions D21 and C6 imply that ."/ ! .0/ as " ! 0. Therefore, ."/  for all " small enough. Thus we get, by D26 , that, for any 0  t ."/ ! 1 as " ! 0 and any A 2 0 , .0/ C  0 ^  00

e

."/ t ."/

De

Q ."/ ; A/  P.t

."/ t ."/ ."/

qQ

.t

Z

t ."/ 0

."/

e

."/ .t ."/ s/

; A/  e .

P ."/ .t ."/  s; A/e 

.0/ C 00 /t ."/

."/ s

FQ ."/ .ds/

.1  FOQ ."/ .t ."/ // ! 0 as " ! 0: (3.3.54)

3.3 Mixed ergodic and large deviation theorems for regenerative processes

141

Relation (3.3.54) implies that, in order to get (3.3.52), we should prove that, for any 0  t ."/ ! 1 as " ! 0 and any A 2 0 , Z t ."/ ."/ ."/ ."/ e  .t s/ P ."/ .t ."/  s; A/e  s FQ ."/ .ds/ ! Q .0/  .0/ .A/ as " ! 0: (3.3.55) 0

Let us also show that, for every A 2 0 , the functions e  are asymptotically bounded, i.e., lim

sup

"!0 0st ."/

e

."/ .t ."/ s/

."/ .t ."/ s/

P ."/ .t ."/  s; A/

P ."/ .t ."/  s; A/ D R < 1:

(3.3.56)

Assume to the contrary that there exists 0  s ."/  t ."/ such that e

."/ .t ."/ s ."/ /

P ."/ .t ."/  s ."/ ; A/ ! 1 as " ! 0:

(3.3.57)

We can suppose that t ."/  s ."/ ! w as " ! 0, where 0  w  1 (if not we can go to subsequences). The limit constant w can not be finite, since in this case, ."/ .t ."/  s ."/ / ! .0/ w as " ! 0 and, therefore, in contradiction with (3.3.57), lim e 

."/ .t ."/ s ."/ /

"!0

P ."/ .t ."/  s ."/ ; A/  lim e 

."/ .t ."/ s ."/ /

"!0

D e

.0/ w

:

(3.3.58)

Neither can w be equal to 1. Indeed, in this case u."/ D t ."/ s ."/ ! 1 as " ! 0. Then, it would follow from (3.3.3) that e

."/ u."/

P ."/ .u."/ ; A/ !  .0/ .A/ as " ! 0;

(3.3.59)

which would again contradict (3.3.57). Therefore, relation (3.3.57) can not hold but, on the contrary, relation (3.3.56) holds. The following cases should be considered: (a) Q .0/ D 0 and (b) Q .0/ 2 .0; 1/. Let us consider case (a). Since ."/ ! .0/ as " ! 0, there exists 0 < 0 <  0 ^  00 such that ."/  .0/ C 0 for all " small enough. This proposition, relation (3.3.56), R T .0/ and inequality 0 e  s FQ ."/ .ds/  Q ."/ ..0/ / imply that Z 1 ."/ ."/ ."/ lim e  .t s/ P ."/ .t ."/  s; A/e  s FQ ."/ .ds/ "!0 0

Z

 lim R "!0

1 0

e

."/ s

 0 Z  lim R e T "!0

C e . D lim Re "!0

T 0

FQ ."/ .ds/ e

0  /T 0

. 0  0 /T

.0/ s

Z Z

1

T 1 0

FQ ."/ .ds/ e .

e .

.0/ C 0 /s

.0/ C 0 /s

 FQ ."/ .ds/

FQ ."/ .ds/ ! 0 as T ! 1:

(3.3.60)

142

3 Nonlinearly perturbed regenerative processes

Consider now case (b). We apply Lemma 1.2.2 for asymptotic analysis of the integral in the right-hand side of (3.3.54). To do this, note first that the integral in (3.3.55) can be rewritten in a form where the limits of integration do not depend on ", Z

t ."/

0

e

."/ .t ."/ s/

Z D

1 0

P ."/ .t ."/  s; A/e 

e

."/ .t ."/ s/

."/ s

FQ ."/ .ds/

P ."/ .t ."/  s; A/e 

."/ s

FQ ."/ .ds/;

(3.3.61)

where we take P ."/ .t  s; A/ D 0 for s > t . Let us show that, for every A 2 0 and any s  0 and v > 0, sup je 

."/ .t ."/ sCu/

P ."/ .t ."/  s C u; A/   .0/ .A/j ! 0 as " ! 0:

(3.3.62)

jujv

Suppose that asymptotic relation (3.3.62) does not take place. This means that there exist sequences 0 < "k ! 0 as k ! 1 and uk with juk j  v such that je 

."/ .t ."/ sCu/

P ."k / .t ."k /  s C uk ; A/   .0/ .A/j ¹ 0 as " ! 0:

(3.3.63)

Since the sequence juk j  v is bounded, .t ."k /  s C uk / ! 1 as k ! 1 for any s  0 and v > 0. Therefore, by (3.3.3), je 

."/ .t ."/ sCu/

P ."k / .t ."k /  s C uk ; A/   .0/ .A/j ! 0 as " ! 0:

(3.3.64)

Relation (3.3.63) contradicts (3.3.64). Thus, relation (3.3.62) holds. Now introduce the distribution functions Z t ."/ e  s FQ ."/ .ds/; t  0: FQQ ."/ .t / D 0

Condition C9 and relation ."/ ! .0/ as " ! 0 imply that there exists 0 < 00 <  0 such that ."/ C 00  .0/ C  0 for all " small enough, and thus Z 1 Z 1 .0/ 0 00 s QQ ."/ lim e F .ds/  lim e . C /s FQ ."/ .ds/ < 1: (3.3.65) 0"!0 0

0"!0 0

Relation (3.3.65) implies that FQQ ."/ .ds/ is a finite measure for " small enough. Let us now assume that condition D20 holds, instead of condition C11 . Condition D20 and relation ."/ ! .0/ as " ! 0 imply that FQQ ."/ ./ ) FQQ .0/ ./ as " ! 0:

(3.3.66)

Relations (3.3.65) and (3.3.66) imply, by Lemma 1.4.4, that FQQ ."/ .1/ D Q ."/ .."/ / ! FQQ .0/ .1/ D Q .0/ ..0/ / as " ! 0:

(3.3.67)

3.3 Mixed ergodic and large deviation theorems for regenerative processes

143

Condition D20 and relations (3.3.56), (3.3.62), and (3.3.66) imply that, for any 0  t ."/ ! 1 as " ! 0 and any A 2 0 , conditions of Lemma 1.2.2 hold for the functions ."/ ."/ e  .t s/ P ."/ .t ."/  s; A/ and the measures FQQ ."/ .ds/. We thus get, by applying Lemma 1.2.2, that for any 0  t ."/ ! 1 as " ! 0 and any A 2 0 , Z 1 ."/ ."/ ."/ e  .t s/ P ."/ .t ."/  s; A/e  s FQ ."/ .ds/ 0

Z D

1

0

Z !

0

1

e

."/ .t ."/ s/

P ."/ .t ."/  s; A/FQQ ."/ .ds/

 .0/ .A/FQQ .0/ .ds/ D Q .0/ ..0/ / .0/ .A/ as " ! 0:

(3.3.68)

Let us consider now the case where condition C11 holds. We show that the proof can be reduced to the case considered above, i.e., to the case where condition D20 holds. Indeed, let us take an arbitrary sequence 0 < "n ! 0 as n ! 1. It is always possible to select from this sequence a subsequence "nk ! 0 as k ! 1 such that FQ ."nk / ./ ) FQ ./ as k ! 1;

(3.3.69)

where FQ .s/ is a proper or improper distribution function. Relation (3.3.69) and the relation ."/ ! .0/ as " ! 0 imply that FQQ ."nk / ./ ) FQQ ./ as k ! 1; where

FQQ .t / D

Z

t 0

e

.0/ s

FQ .ds/;

(3.3.70)

t  0:

The proof given above for the case where condition D20 holds can be applied to a subsequence of models with perturbation parameters "nk , k  1, and to yield, for any A 2 0 , the following relation: Z 1 Q .0/ / .0/ .A/ as k ! 1; (3.3.71) P ."nk / .t ."nk /  s; A/FQQ ."nk / .ds/ ! . 0

where Q .0/ / D .

Z

t 0

e

.0/ s

FQ .ds/:

The limit distribution FQ .s/ does depend on the choice of the sequence "n and the Q .0/ / does not. subsequence "nk , but the limit . Indeed, relations (3.3.65) and (3.3.70), by Lemma 3.3.1, imply that Q .0/ / as k ! 1: Q ."nk / .."nk / / ! .

(3.3.72)

144

3 Nonlinearly perturbed regenerative processes

On the other hand, by condition C11 , Q ."nk / ..0/ / ! Q .0/ as k ! 1:

(3.3.73)

Q ."nk / .."nk / /  Q ."nk / ..0/ / ! 0 as k ! 1:

(3.3.74)

Let us now show that

Indeed, using condition C9 , the relation ."nk / ! .0/ as k ! 1, and the inequality ."/ C 00  .0/ C  0 , which holds for all " small enough, we get lim j Q ."nk / .."nk / /  Q ."nk / ..0/ / j

k!1

 lim . sup j e 

."nk /

s

k!1 0sT

Z

C 0

1 T

e 0

."nk / s

 e

.0/ s

j

FQ ."nk / .ds/ C

 .e  0 T C e  T / lim

Z

k!1 0

1

e .

Z

1 T

.0/ C 0 /s

e

.0/ s

FQ ."nk / .ds//

FQ ."nk / .ds/ ! 0 as T ! 1: (3.3.75)

Q .0/ / D Q .0/ and, thereRelations (3.3.72), (3.3.73), and (3.3.74) imply that . fore, the limit in the right-hand side of (3.3.71) does not depend on the choice of the sequence "n and the subsequence "nk . Moreover, it coincides with the limit in the right-hand side of relation (3.3.68). This implies that relation (3.3.68) holds. The proof of the theorem is complete. There is a difference between Theorems 3.3.8 and 3.3.9. In Theorem 3.3.8, the limit in the asymptotic relation (3.3.28) depends on characteristics of the transition period via the quantity 1  fQ.0/ D lim"!0 FQ ."/ .1/ while, in Theorem 3.3.9, the limit in the asymptotic relation (3.3.52) depends on characteristics R 1 .0/ of the transition period via the quantity Q .0/ D lim"!0 0 e  s FQ ."/ .ds/. Obviously, 1  fQ.0/  Q .0/ . Let us consider the case where condition D20 holds, i.e., FQ ."/ ./ ) FQ .0/ ./ as " ! 0. In this case, under an additional assumption that condition C9 holds, we have R 1 .0/ 1  fQ.0/ D FQ .0/ .1/ and Q .0/ D Q .0/ ..0/ / D 0 e  s FQ .0/ .ds/. If .0/ D 0, then Q .0/ D 1  fQ.0/ . In this case, Theorem 3.3.9 just reduces to Theorem 3.3.8. The limits in (3.3.28) and (3.3.52) do not depend on the characteristics of the transition period if 1  fQ.0/ D 1. If .0/ > 0 and FQ .0/ .0/ < 1, i.e., the distribution function FQ .0/ .t / is not concentrated in zero, then FQ .0/ .1/ < Q .0/ ..0/ /. The limit in (3.3.52) does not depend on the characteristics of the transition period if Q .0/ ..0/ / D 1. The following theorem, which can be referred as a quasi-ergodic theorem, is a direct corollary of Theorem 3.3.9.

145

3.3 Mixed ergodic and large deviation theorems for regenerative processes

Theorem 3.3.10. Let conditions D21 , C6 , F7 , C9 , D26 , and C11 , together with the additional assumption that Q .0/ > 0, hold. Then, for any 0  t ."/ ! 1 as " ! 0 and any A 2 0 , Pf ."/ .t ."/ / 2 A=."/ > t ."/ g D !

Pf ."/ .t ."/ / 2 A; ."/ > t ."/ g Pf."/ > t ."/ g Q .0/ .A/ D  .0/ .A/ as " ! 0: Q .0/ .X/

(3.3.76)

Thus, convergence of the conditional distribution of a perturbed regenerative process to the corresponding quasi-stationary distribution takes place also in a model with transition period. Note that Theorem 3.3.9 yields directly that the conditional probabilities in (3.3.76) converge to the fraction Q .0/ Q .0/ .A/=Q .0/ Q .0/ .X/ as " ! 0. In order to be able to reduce this expression to Q .0/ .A/=Q .0/ .X/ D  .0/ .A/, one should require that Q .0/ > 0. Without this assumption, the asymptotic relation (3.3.52) would still take place, but the asymptotic relation (3.3.76) possibly would not.

3.3.8 Quasi-stationary distributions for perturbed regenerative processes with transition period Let "1 , "2 , and "3 be parameters used, respectively, in conditions D23 , C6 , and F9 . R 1 .0/ Condition C10 implies that 0 e . C /s FOQ ."/ .ds/ < 1 for all " small enough, say "  "4 . Define "0 D min."1 ; "2 ; "3 ; "4 /;

(3.3.77)

and denote QQ ."/ .A/ D Q ."/ .."/ /Q ."/ .A/;

A 2 :

(3.3.78)

The following theorem is a direct corollary of Theorems 3.3.3 and 3.3.9. Theorem 3.3.11. Let conditions D23 , C6 , F9 , and C10 hold. Then, for "  "0 , e

."/ t

Pf ."/ .t / 2 A; ."/ > t g ! QQ ."/ .A/ as t ! 1; A 2 0 :

(3.3.79)

Let us assume the following condition: D27 : FQ ."/ .1/ > 0 for "  "5 . Condition D27 obviously implies that Q ."/ .."/ / > 0 for "  "5 . Let us also define "00 D min."0 ; "5 /: Theorem 3.3.11 implies the following statement.

(3.3.80)

146

3 Nonlinearly perturbed regenerative processes

Theorem 3.3.12. Let conditions D23 , C6 , F9 , C10 , and D27 hold. Then, for "  "00 , Pf ."/ .t / 2 A=."/ > t g D !

Pf ."/ .t / 2 A; ."/ > t g Pf."/ > t g Q ."/ .A/ D  ."/ .A/ as t ! 1; A 2 0 : Q ."/ .X/

(3.3.81)

Finally, we also give conditions for convergence of the functionals QQ ."/ .A/. In this case, it is natural to assume also that condition D20 holds. Theorem 3.3.13. Let conditions D21 , C6 , F7 , and D20 , C9 hold. Then QQ ."/ .A/ ! QQ .0/ .A/ as " ! 0;

A 2 0 :

(3.3.82)

Proof. According to (3.3.13), conditions D21, C6 , and F7 imply that Q ."/ .A/ ! Q .0/ .A/ as " ! 0 for A 2 0 . Thus, it is sufficient to show that conditions D24 and C9 imply that Q ."/ .."/ / ! Q .0/ ..0/ / as " ! 0:

(3.3.83)

Since ."/ ! .0/ as " ! 0, we have ."/ < ˇ < .0/ C  0 for all " small enough. Thus, Z 1 ."/ e s FQ ."/ .ds/ lim lim T !1 "!0 T

Z

 lim lim

1

T !1 "!0 T

 lim e T . T !1

e sˇ FQ ."/ .ds/

.0/ C 0 ˇ /

Z lim

"!0 0

1

e s.

.0/ C 0 /

FQ ."/ .ds/ D 0:

(3.3.84)

Condition D24 and the relation ."/ ! .0/ as " ! 0 imply that, for any T which is a point of continuity of the distribution FQ .0/ .s/, Z T Z T .0/ s."/ Q ."/ lim e e s FQ .0/ .ds/: (3.3.85) F .ds/ D "!0 0

0

Relations (3.3.84) and (3.3.85) imply relation (3.3.83).

3.4 Exponential expansions for nonlinearly perturbed regenerative processes In this section, we present exponential asymptotic expansions in mixed ergodic and large deviation theorems for nonlinearly perturbed regenerative processes and regenerative stopping times.

3.4

Exponential expansions for nonlinearly perturbed regenerative processes

147

3.4.1 Pseudo-stationary exponential expansions for perturbed regenerative processes based on stopping probabilities Let us now formulate an analogue of Theorem 2.1.1. We assume the following balancing condition: .k/

B4 : 0  t ."/ ! 1 and f ."/ ! 0 as " ! 0 in such a way that .f ."/ /k t ."/ ! k , where 0  k < 1. ."/

Obviously, conditions D13 and C3 imply that the power moments mr are finite for all r  1 and all " small enough, and that .0/ m."/ r ! mr < 1 as " ! 0;

r  1:

(3.4.1)

Theorem 3.4.1. Let conditions D13 and C3 hold. Then (i) The following asymptotic Taylor’s expansion holds for the root ."/ of the equation (3.3.2) for every k  1: ."/ D b"1 f ."/ C    C b"k .f ."/ /k C o..f ."/ /k /;

(3.4.2)

where the coefficients b"k , k  1, can be calculated using the recurrence for."/ 1 ."/ 2 1 1 mulas b"1 D .m."/ 1 / , b"2 D  2 .m1 / m2 b"1 and, in general, for k  2, b"k D

1  .m."/ 1 /

k X r D2

m."/ r



X

k1 Y

ni b"i =ni Š;

(3.4.3)

n1 ;:::;nk1 2Dr;k iD1

where Dr;k , for every 2  r  k < 1, is the set of all nonnegative, integer solutions of the system n1 C    C nk1 D r; n1 C    C .k  1/nk1 D k:

(3.4.4)

(ii) If, additionally, conditions B4.k/ , for some k  1, and F7 hold, then the following asymptotic relation holds for any A 2 0 : Pf ."/ .t ."/ / 2 A; ."/ > t ."/ g expf.b"1 f ."/ C    C b"k1 .f ."/ /k1 /t ."/ g !  .0/ .A/e k b0k as " ! 0:

(3.4.5)

Proof. The proof can be obtained by directly applying Theorem 2.1.1 to the renewal equation (3.2.1). Condition D13 implies that condition D6 holds for the distribution functions F ."/ .s/. Also, condition C3 implies condition C1 . Condition F7 .a/ implies that, for every A 2 0 , condition F4 .a/ holds for the function q ."/ .s; A/. Since 0  q ."/ .s; A/  1, condition F5 .b/ obviously holds. As was shown in the proof of Theorem 3.3.1, condition C3 implies that, for every A 2 0 , condition F5 .c/ holds. Finally, condition .k/ .k/ B4 coincides in this case with condition B2 . Thus, all conditions of Theorem 2.1.1 hold.

148

3 Nonlinearly perturbed regenerative processes

3.4.2 Pseudo-stationary exponential expansions for perturbed regenerative processes in the case where the stopping probabilities are of order O."/ Let us now assume the following perturbation condition: .k/

P9 :

(a) 1  f ."/ D 1 C b1;0 " C    C bk;0 "k C o."k /, where jbr;0 j < 1, r D 1; : : : ; k; ."/

.0/

(b) mr D mr C b1;r " C    C bkr;r "kr C o."kr / for r D 1; : : : ; k, where jbl;r j < 1, l D 1; : : : ; k  r, r D 1; : : : ; k. .0/

It is convenient to also set b0;0 D 1 and b0;r D mr , r  1. We shall also use the following balancing condition for 1  r  k: .r/

B5 : 0  t ."/ ! 1 as " ! 0 such that "r t ."/ ! r , where 0  r < 1. Theorem 3.4.2. Let conditions D13 , C3 and P9.k/ hold. Then (i) For all " small enough there exists a unique nonnegative root ."/ of the equation (3.3.2) with the following asymptotic Taylor’s expansion: ."/ D a1 " C    C ak "k C o."k /;

(3.4.6)

where the coefficients an , n D 1; : : : ; k, are given by the recurrence formulas a1 D b1;0 =b0;1 and, in general, for n D 1; : : : ; k,  n1 X 1 bn;0 C bnq;1 aq an D b0;1 qD1

C

n X n X mD2 qDm

bnq;m 

X

q1 Y

 n app =np Š ;

(3.4.7)

n1 ;:::;nq1 2Dm;q pD1

where Dm;q , for every 2  m  q < 1, is the set of all nonnegative, integer solutions of the system n1 C    C nq1 D m; n1 C    C .q  1/nq1 D q:

(3.4.8)

(ii) If bl;0 D 0, l D 1; : : : ; r, for some 1  r  k, then a1 ; : : : ; ar D 0. If bl;0 D 0, l D 1; : : : ; r  1, but br;0 < 0 for some 1  r  k, then a1 ; : : : ; ar1 D 0 but ar > 0. (iii) If, additionally, conditions B.r/ 5 for some 1  r  k, and F7 hold, then the following asymptotic relation holds for any A 2 0 : Pf ."/ .t ."/ / 2 A; ."/ > t ."/ g !  .0/ .A/e r ar as " ! 0: expf.a1 " C    C ar 1 "r 1 /t ."/ g

(3.4.9)

3.4

Exponential expansions for nonlinearly perturbed regenerative processes

149

Proof. The proof can be obtained by a direct application of Theorem 2.1.2 to the renewal equation (3.2.1). Condition D13 implies that condition D6 holds for the distribution functions F ."/ .s/. Also, condition C3 implies condition C1 . Condition F7 .a/ implies that, for every A 2 0 , condition F5 .a/ holds for the function q ."/ .s; A/. Since 0  q ."/ .s; A/  1, condition F5 .b/ obviously holds. As was shown in the proof of Theorem 3.3.1, condition C3 implies that for every A 2 0 condition F5 .c/ is fulfilled. Finally, condition .r/ .r/ .k/ B5 coincides in this case with condition B3 , while condition P9 coincides with .k/ condition P1 . Thus, all conditions of Theorem 2.1.2 hold.

3.4.3 Pseudo-stationary exponential expansions for perturbed regenerative processes in the case where the stopping probabilities are of order O."h / Let us introduce integer parameters 1  h  k0 and kr  1, r D 1; : : :, and define a vector parameter kN D .h; k0 ; : : : ; kNQ /, where NQ D min.n W nh  k0 /. We assume to hold the following perturbation condition: N

.k/ P10 :

(a) 1  f ."/ D 1 C bh;0 "h C    C bk0 ;0 "k0 C o."k0 /, where jbl;0 j < 1, l D h; : : : ; k0 ; .0/ kr C o."kr / for r D 1; : : : ; N Q , where (b) m."/ r D mr C b1;r " C    C bkr ;r " Q jbl;r j < 1; l D 1; : : : ; kr , r D 1; : : : ; N . .0/

As before, it is convenient to also define b0;0 D 1 and b0;r D mr , r  1. .kN /

.k/

Condition P10 is more general than condition P9 , and the first one reduces to the second one if h D 1, k0 D k (in this case, NQ D k) and kr D k  r, r D 1; : : : ; NQ . The following theorem supplements and generalises Theorem 3.4.2. .kN /

Theorem 3.4.3. Let conditions D13 , C3 and P10 hold. Then (i) For all " small enough there exists a unique nonnegative root ."/ of the equation (3.3.2) with the following asymptotic expansion: ."/ D ah "h C    C ak "k C o."k /;

(3.4.10)

where k D min.k0 ; k1 Ch; : : : ; kNQ ChNQ / and the coefficients an , n D h; : : : ; k, are given by the recurrence formulas ah D bh;0 =b0;1 and for n D h; : : : ; k, an D

1 b0;1

 n1 X bn;0 C bni;1 ai iDh

C

Œn= Xh

n X

r D2 j Dr h

bnj;r 

X

jY 1

nh ;:::;nj 1 2Dh;r;j iDh

 ; (3.4.11)

aini =ni Š

150

3 Nonlinearly perturbed regenerative processes

where Dh;r;j , for every h  1 and 2  r  j < 1, is the set of all nonnegative, integer solutions of the system nh C    C nj 1 D r; hnh C    C .j  1/nj 1 D j:

(3.4.12)

(ii) If bl;0 D 0, l D h; : : : ; r, for some h  r  k, then a1 ; : : : ; ar D 0. If bl;0 D 0, l D h; : : : ; r  1 but br;0 < 0 for some h  r  k, then ah ; : : : ; ar1 D 0 but ar > 0. (iii) If, additionally, conditions B.r/ 5 , for some h  r  k, and F7 hold, then the following asymptotic relation holds: Pf ."/ .t ."/ / 2 A; ."/ > t ."/ g !  .0/ .A/e r ar as " ! 0: (3.4.13) expf.ah " C    C ar 1 "r 1 /t ."/ g Proof. The proof can be obtained by a direct application of Theorem 2.1.3 to the renewal equation (3.2.1). Condition D13 implies that condition D6 holds for the distribution functions F ."/ .s/. Also, condition C3 implies condition C1 . Condition F7 .a/ implies that, for every A 2 0 , condition F5 .a/ holds for the function q ."/ .s; A/. Since 0  q ."/ .s; A/  1, condition F5 .b/ obviously holds. As was shown in the proof of Theorem 3.3.1, condition C3 implies that for every A 2 0 condition F5 .c/ .r/ .r/ holds true. Finally, condition B5 coincides in this case with condition B3 , while .k/ .k/ condition P9 coincides with condition P1 . Thus, all conditions of Theorem 2.1.3 are satisfied.

3.4.4 Quasi-stationary exponential expansions for perturbed regenerative processes in the case where the perturbation is of order O."/ We finally consider the most general case where the limit distribution F .0/ .s/ can be proper or improper. Let us introduce the following moment generating functions: Z 1  ."/ .; r/ D s r e s F ."/ .ds/; r  1: 0

Condition C3 implies that for any ˇ < ı the mixed moment generating functions satisfy  ."/ .ˇ; r/ < 1; r D 0; 1; : : :, for all " small enough. Let us introduce the following perturbation condition: .k/

P11 :  ."/ ..0/ ; r/ D  .0/ ..0/ ; r/ C b1;r " C    C bkr;r "kr C o."kr / for r D 0; : : : ; k, where jbn;r j < 1, n D 1; : : : ; k  r, r D 0; : : : ; k. It is convenient to define b0;r D  .0/ ..0/ ; r/, r D 0; 1; : : : . It clearly follows from the definition of .0/ that b0;0 D  .0/ ..0/ ; 0/ D 1.

3.4

Exponential expansions for nonlinearly perturbed regenerative processes

151

Let us formulate an analogue of Theorem 2.2.1. .k/

Theorem 3.4.4. Let conditions D21 , C6 and P11 hold. Then (i) The root ."/ of equation (3.3.2) has the asymptotic expansion ."/ D .0/ C a1 " C    C ak "k C o."k /;

(3.4.14)

where the coefficients an are given by the recurrence formulas a1 D b1;0 =b0;1 and, in general for n D 1; : : : ; k, an D

1 b0;1

 bn;0 C

n1 X

bnq;1 aq

qD1

C

X

n X

bnq;m 

2mn qDm

X

q1 Y



n app =np Š

; (3.4.15)

n1 ;:::;nq1 2Dm;q pD1

were Dm;q , for every 2  m  q < 1, is the set of all nonnegative, integer solutions of the system n1 C    C nq1 D m; n1 C    C .q  1/nq1 D q:

(3.4.16)

(ii) If the coefficients satisfy bl;0 D 0, l D 1; : : : ; r, for some 1  r  k, then a1 ; : : : ; ar D 0. If bl;0 D 0, l D 1; : : : ; r  1, but br;0 < 0 for some 1  r  k, then a1 ; : : : ; ar 1 D 0 but ar > 0. .r/

(iii) If, additionally, conditions B5 , for some 1  r  k, and F7 hold, then for any A 2 0 , the following asymptotic relation holds: Pf ."/ .t ."/ / 2 A; ."/ > t ."/ g expf..0/ C a1 " C    C ar1 "r1 /t ."/ g ! Q .0/ .A/e r ar as " ! 0:

(3.4.17)

Proof. The proof can be obtained by directly applying Theorem 2.2.1 to the renewal equation (3.2.1). Condition D21 implies that condition D11 holds for the distribution functions F ."/ .s/. Also, condition C6 implies condition C2 . Condition F7 .a/ implies that for every A 2 0 condition F6 .a/ holds for the function q ."/ .s; A/. Since 0  q ."/ .s; A/  1, condition F6 .b/ obviously holds. As was shown in the proof of Theorem 3.3.1, condition C6 implies that for every A 2 0 , condition F6 .c/ is verified. .r/ .r/ .k/ Finally, condition B5 coincides in this case with condition B3 , while condition P11 .k/ coincides with condition P3 . Thus, all conditions of Theorem 2.2.1 hold.

152

3 Nonlinearly perturbed regenerative processes

3.4.5 Quasi-stationary exponential expansions for perturbed regenerative processes in the case where the perturbation is of order O."h / Let us also formulate a theorem which is an analogue of Theorem 3.4.3 for the model of improper perturbed regenerative processes. As above, let us introduce the integer parameters 1  h  k0 and kr  1, r D 1; : : :, and define a vector parameter kN D .h; k0 ; : : : ; kNQ /, where NQ D min.n W nh  k0 /. We assume that the following perturbation condition holds: N

.k/ : P12

(a)  ."/ ..0/ ; 0/ D  .0/ ..0/ ; 0/ C bh;0 "h C    C bk0 ;0 "k0 C o."k0 /, where jbn;0 j < 1; n D h; : : : ; k0 ; (b)  ."/ ..0/ ; r/ D  .0/ ..0/ ; r/ C b1;r " C    C bkr ;r "kr C o."kr / for r D 1; : : : ; k, where jbn;r j < 1, n D 1; : : : ; kr , r D 1; : : : ; NQ .

It is convenient to define b0;r D  .0/ ..0/ ; r/, r D 0; 1; : : : . Note that  .0/ ..0/ ; 0/ D 1, while  .0/ ..0/ ; r/ > 0, r D 1; : : : ; NQ . .kN / .k/ Condition P12 is more general than condition P11 , and the first one reduces to the Q second one if h D 1, k0 D k (in this case, N D k) and kr D k  r, r D 1; : : : ; N . The following theorem generalises Theorems 3.4.4. N

.k/ Theorem 3.4.5. Let conditions D21 , C6 and P12 hold. Then

(i) For all " small enough there exists a unique nonnegative root ."/ of equation (2.1.4) with the following asymptotic expansion: ."/ D .0/ C ah "h C    C ak "k C o."k /;

(3.4.18)

where k D min.k0 ; k1 Ch; : : : ; kNQ ChNQ / and the coefficients an , n D h; : : : ; k, are given by the recurrence formulas ah D bh;0 =b0;1 and for n D h; : : : ; k, an D

1 b0;1

 bn;0 C

n1 X

bni;1 ai

iDh

C

Œn= Xh

n X

r D2 j Dr h

bnj;r 

X

jY 1

 aini =ni Š ;

(3.4.19)

nh ;:::;nj 1 2Dh;r;j iDh

where Dh;r;j , for every h  1 and 2  r  j < 1, is the set of all nonnegative, integer solutions of the system nh C    C nj 1 D r; hnh C    C .j  1/nj 1 D j:

(3.4.20)

(ii) If bl;0 D 0, l D h; : : : ; r, for some h  r  k, then a1 ; : : : ; ar D 0. If bl;0 D 0, l D h; : : : ; r  1 but br;0 < 0 for some h  r  k, then ah ; : : : ; ar1 D 0 but ar > 0.

3.4

Exponential expansions for nonlinearly perturbed regenerative processes

153

.r/

(iii) If, additionally, conditions B5 , for some h  r  k, and F7 hold, then the following asymptotic relation holds: Pf ."/ .t ."/ / 2 A; ."/ > t ."/ g expf..0/ C ah "h C    C ar1 "r1 /t ."/ g ! Q .0/ .A/e r ar as " ! 0:

(3.4.21)

Proof. The proof can be obtained by applying Theorem 2.2.2 to the renewal equation (3.2.1). Condition D21 implies that condition D11 holds for the distribution functions F ."/ .s/. Also, condition C6 implies condition C2 . Condition F7 .a/ implies that, for every A 2 0 , condition F6 .a/ holds for the function q ."/ .s; A/. Since 0  q ."/ .s; A/  1, condition F6 .b/ obviously holds. As was shown in the proof of Theorem 3.3.1, condition C6 implies that, for every A 2 0 , condition F6 .c/ also holds. .r/ .k/ Finally, condition B.r/ 5 coincides with condition B3 , and condition P12 with condition P4.k/ . This shows that all conditions of Theorem 2.2.2 hold.

3.4.6 Pseudo-stationary and quasi-stationary exponential expansions for perturbed regenerative processes with transition period Let us also give the corresponding exponential expansions for perturbed regenerative processes with transition period. The exponential expansions presented above in Theorems 3.4.1, 3.4.2, 3.4.3, 3.4.4, and 3.4.5 are based on combination of two asymptotic results. The first one is a mixed ergodic and large deviation asymptotic relation of the form Pf ."/ .t ."/ / 2 A; ."/ > t ."/ g ! Q .0/ .A/ as " ! 0: expf."/ t ."/ g

(3.4.22)

The second one is a representation of the characteristic root ."/ of equation (2.1.4) in the form of the asymptotic expansion ."/ D .0/ C ah "h C    C ak "k C o."k /:

(3.4.23)

Representation (3.4.23) and the additional balancing condition, "r t ."/ ! r as " ! 0, permit to transform asymptotic relation (3.4.22) to an equivalent but more explicit form, Pf ."/ .t ."/ / 2 A; ."/ > t ."/ g ! Q .0/ .A/e r ar as " ! 0: (3.4.24) expf..0/ C ah "h C    C ar 1 "r 1 /t ."/ g In the model of asymptotically proper perturbed regenerative processes (f ."/ ! f D 0 as " ! 0), the mixed ergodic and large deviation asymptotic relation (3.4.22) .0/

154

3 Nonlinearly perturbed regenerative processes

takes the same form for the perturbed regenerative processes with transition period as for the corresponding standard regenerative processes. In this case, the asymptotic relation (3.4.24) is also preserved, for processes with transition period, in the same form as for the corresponding standard regenerative processes. However, the corresponding conditions imposed on characteristics of the transition period should be augmented. The next three theorems are related to the model of asymptotically proper perturbed regenerative processes with transition period. Theorem 3.4.6. Let conditions D13 , C3 , F7 , D17 , C7 , D25 , and B4.k/ , for some k  1, hold. Then, for any A 2 0 , Pf ."/ .t ."/ / 2 A; ."/ > t ."/ g expf.b"1 f ."/ C    C b"k1 .f ."/ /k1 /t ."/ g ! .1  f .0/ / .0/ .A/e k b0k as " ! 0: .k/

(3.4.25) .r/

Theorem 3.4.7. Let conditions D13 , C3 , F7 , D17 , C7 , D25 , P9 , and B5 , for some 1  r  k, hold. Then, for any A 2 0 , Pf ."/ .t ."/ / 2 A; ."/ > t ."/ g expf.a1 " C    C ar 1 "r1 /t ."/ g ! .1  f .0/ / .0/ .A/e r ar as " ! 0: .k/

(3.4.26) .r/

Theorem 3.4.8. Let conditions D13 , C3 , F7 , D17 , C7 , D25 , P10 , and B5 , for some 1  r  k, hold. Then, for any A 2 0 , Pf ."/ .t ."/ / 2 A; ."/ > t ."/ g expf.ah " C    C ar 1 "r1 /t ."/ g ! .1  f .0/ / .0/ .A/e r ar as " ! 0:

(3.4.27)

Slightly different is the situation for the model of asymptotically improper perturbed regenerative processes (f ."/ ! f .0/  0 as " ! 0). The mixed ergodic and large deviation asymptotic relation takes the form (3.4.22) for standard regenerative processes, while, for the perturbed regenerative processes with transition period, it takes a similar but different form, Pf ."/ .t ."/ / 2 A; ."/ > t ."/ g ! Q .0/ ..0/ /Q .0/ .A/ as " ! 0: expf."/ t ."/ g

(3.4.28)

Representation (3.4.23) plus an additional balancing condition, "r t ."/ ! r as " ! 0, allow to transform the asymptotic relation (3.4.28) to an equivalent but more

155

3.5 Asymptotic expansions for quasi-stationary distributions

explicit form, Pf ."/ .t ."/ / 2 A; ."/ > t ."/ g expf..0/ C ah "h C    C ar1 "r1 /t ."/ g ! Q .0/ ..0/ /Q .0/ .A/e r ar as " ! 0:

(3.4.29)

Again, the corresponding conditions imposed on characteristics of the transition period should augmented. The next two theorems relate to the model of asymptotically improper perturbed regenerative processes with transition period. It should be noted that these theorems cover, in fact, both the cases where f .0/ > 0 and f .0/ D 0. In the latter case, these theorems reduce, respectively, to Theorems 3.4.7 and 3.4.8. .k/

.r/

Theorem 3.4.9. Let conditions D21 , C6 , F7 , C9 , D26 , C11 , P11 and B5 , for some h  r  k, hold. Then, for any A 2 0 , Pf ."/ .t ."/ / 2 A; ."/ > t ."/ g expf..0/ C a1 " C    C ar1 "r1 /t ."/ g ! Q .0/ Q .0/ .A/e r ar as " ! 0:

(3.4.30)

.k/ Theorem 3.4.10. Let conditions D21 , C6 , F7 , C9 , D26 , C11, P12 and B.r/ 5 , for some h  r  k, hold. Then, for any A 2 0 ,

Pf ."/ .t ."/ / 2 A; ."/ > t ."/ g expf..0/ C ah "h C    C ar1 "r1 /t ."/ g ! Q .0/ Q .0/ .A/e r ar as " ! 0:

(3.4.31)

3.5 Asymptotic expansions for quasi-stationary distributions In this section, we give asymptotic expansions for stationary and quasi-stationary distributions of perturbed regenerative processes.

3.5.1 Asymptotic expansions for stationary distributions of perturbed regenerative processes Let us define, for r D 0; 1; : : :, moment functionals by Z 1 s r q ."/ .s; A/m.ds/; qr."/ .A/ D 0

A 2 :

156

3 Nonlinearly perturbed regenerative processes

Let us first consider the simplest case where condition D14 holds, i.e., f ."/ D 0 for every "  "0 . As was shown in Lemma 3.1.2, this condition holds if and only if the regenerative stopping time ."/ D 1 with probability 1 for all "  "0 . ."/ ."/ ."/ In this case, obviously, F ."/ .s/ D FO ."/ .s/ D Pf 1  sg and m1 D E 1 . Also, ."/ q ."/ .s; A/ D Pf ."/ .t / 2 A; 1 > t g. The following formula defines a stationary distribution for the regenerative process ."/ .t /, t  0: R1 

."/

.A/ D

0

q ."/ .s; A/m.ds/ m."/ 1

;

A 2 :

(3.5.1)

Let us assume the following two perturbation conditions: .k/ .0/ .0/ k k : m."/ P13 1 D m1 C b1;1 " C    C bk;1 " C o." /, where m1 2 .0; 1/ and jbn;1 j < 1, n D 1; : : : ; k;

and .k/

."/

.0/

P14 : q0 .A/ D q0 .A/Cc1;0 .A/"C   Cck;0 .A/"k Co."k /, where jcn;0 .A/j < 1, n D 1; : : : ; k, A 2 0 . .0/

.0/

Note that m1 and q0 .A/ in the conditions above are just some constants that may or may not be the corresponding characteristics of the limit regenerative process. The .0/ condition m.0/ 1 2 .0; 1/ is essential. As far as the functional q0 .A/ is concerned, it ."/ automatically takes values in the interval Œ0; 1, since the pre-limit functional q0 .A/ takes values in this interval. .0/ .0/ It is also convenient to define b0;1 D m1 and c0;0 .A/ D q0 .A/. .k/ .k/ Theorem 3.5.1. Let conditions P13 and P14 hold. Then, for any A 2 0 , the following asymptotic expansions hold:

 ."/ .A/ D

.0/

q0 .A/ C c1;0 .A/" C    C ck;0 .A/"k C o."k / k k m.0/ 1 C b1;1 " C    C bk;1 " C o." /

D  .0/ .A/ C f1 .A/" C    C fk .A/"k C o."k /;

(3.5.2)

where the coefficients fn .A/ are given by the recurrence formulas f0 .A/ D  .0/ .A/ D c0;0 .A/=b0;1 and, for n D 0; : : : ; k,   n1 X bnq;1 fq .A/ =b0;1 : fn .A/ D cn;0 .A/  qD0

(3.5.3)

157

3.5 Asymptotic expansions for quasi-stationary distributions

Proof. The theorem directly follows from Theorem 2.3.1 that should be applied to any ."/ Q ."/ .1/ D  ."/ .A/ D q0."/ .A/=m."/ A 2 0 with q ."/ D q0."/ .A/, m."/ 1 D E 1 and x 1 . .k/ .k/ .k/ .k/ In this case, conditions P13 and P14 reduce, respectively, to conditions P3 and P4 . Alternatively, one can directly transform the rational expansion (3.5.2) given by for.k/ .k/ and P14 to a polynomial form. Note that b0;1 ¤ 0. mula (3.5.1) and conditions P13

3.5.2 Asymptotic expansions for quasi-stationary distributions of perturbed regenerative processes Let us consider the general case. We define, for r D 0; 1; : : :, the moment functionals Z 1 ! ."/ .; r; A/ D s r e s q ."/ .s; A/m.ds/;  2 R1 ; A 2 : 0

It follows from conditions D21 , C6 , and F7 that for any 0  ˇ < ı we have ! ."/ .ˇ; r; A/ < 1 with r D 0; 1; : : : for all " small enough. Indeed, let cr D sups0 s r e .ıˇ /s . Then, for r D 0; 1; : : :, Z 1 ."/ s r e ˇs q ."/ .s; A/m.ds/ ! .ˇ; r; A/ D Z 

0 1

0

s r e ˇs .1  FO ."/ .s// ds  cr

 cr ı 1 .1 C

Z

1 0

Z

1

0

e ıs .1  FO ."/ .s// ds

e ıs FO ."/ .ds// < 1:

(3.5.4)

We now formulate a perturbation condition for the forcing functions q ."/ .t; A/: .k/ P15 : ! ."/ ..0/ ; r; A/ D ! .0/ ..0/ ; r; A/ C c1;r .A/" C    C ckr;r .A/"kr C o."kr / for r D 0; : : : ; k, where jcn;r j < 1, n D 1; : : : ; k  r, r D 0; : : : ; k; A 2 0 .

It is convenient to define c0;r .A/ D ! .0/ ..0/ ; r; A/, r D 0; 1; : : : . It is useful to note that in the case where f .0/ D 0, the characteristic root is ."/ .0/ D 0 and, therefore, the mixed moment functionals ! ."/ .0; r; A/ D qr .A/ D R 1 r ."/ R 1 r ."/ ."/ ."/ D 0 s F .ds/ are the usual mo0 s q .s; A/m.ds/ and  .0; r/ D mr ment functionals. .k/ In this case, perturbation condition P11 reduces to the perturbation condition Pk9 . .k/ The perturbation condition P15 also reduces to a simpler form: .k/

."/

.0/

P16 : qr .A/ D qr .A/Cc1;r .A/"C  Cckr;r .A/"kr Co."kr / for r D 0; : : : ; k, where jcn;r .A/j < 1, n D 1; : : : ; k  r, r D 0; : : : ; k, A 2 0 . Recall that Q ."/ .A/ D

! ."/ .."/ ; 0; A/ ;  ."/ .."/ ; 1/

A 2 :

(3.5.5)

158

3 Nonlinearly perturbed regenerative processes .kC1/

Theorem 3.5.2. Let conditions D21 , C6 , F7 , P11 A 2 0 , the following asymptotic expansions hold: Q ."/ .A/ D

.k/

, and P15 hold. Then, for any

! .0/ ..0/ ; 0; A/ C f10 .A/" C    C fk0 .A/"k C o."k /  .0/ ..0/ ; 1/ C f100 " C    C fk00 "k C o."k /

D Q .0/ .A/ C f1 .A/" C    C fk .A/"k C o."k /;

(3.5.6)

where the coefficients fn0 .A/; fn00 are given by the formulas f00 .A/ D ! .0/ ..0/ ; 0; A/ D c0;0 .A/, f10 .A/ D c1;0 .A/ C c0;1 .A/a1 , f000 D  .0/ ..0/ ; 1/ D b0;1 , f100 D b1;1 C b0;2 a1 , and, for n D 0; : : : ; k, fn0 .A/ D cn;0 .A/ C

n X

cnq;1 .A/aq

qD1

X

C

n X

X

cnq;m .A/ 

2mn qDm

q1 Y

n

app =np Š;

(3.5.7)

n1 ;:::;nq1 2Dm;q pD1

and fn00 D bn1 C

n X

bnq;2 aq

qD1

C

X

n X

2mn qDm

X

bnq;mC1 

q1 Y

n

app =np Š;

(3.5.8)

n1 ;:::;nq1 2Dm;q pD1

and the coefficients fn .A/ are given by the recurrence formulas f0 .A/ D Q .0/ .A/ D f00 .A/=f000 and, for n D 0; : : : ; k,   n1 X 0 00 fnq fq .A/ =f000 : fn .A/ D fn .A/ 

(3.5.9)

qD0

Proof. The theorem directly follows from Theorem 2.3.2 that should be applied, for ."/ any A 2 0 , to a model in which q ."/ .s/ D q ."/ .s; A/, F ."/ .s/ D Pf 1  s; ."/ >

1."/ g and xQ ."/ .1/ D Q ."/ .A/ D ! ."/ .."/ ; 0; A/= ."/ .."/ ; 1/. In this case, as was shown in the proof of Theorem 3.4.4, conditions D21 , C6 and F7 .kC1/ .k/ imply that conditions D13, C2 and F6 hold. Also, conditions P11 and P15 coincide with conditions P3.kC1/ and P7.k/ , respectively. Recall that QQ ."/ .A/ D Q ."/ .."/ ; 0/Q ."/ .A/;

A 2 ;

(3.5.10)

159

3.5 Asymptotic expansions for quasi-stationary distributions

where Q ."/ .; r/ D

Z

1

0

s r e s FQ ."/ .ds/;

 2 R1 ;

r D 0; 1; : : : :

Note that, by the definition, Q ."/ .; 0/ D Q ."/ ./. Condition C9 implies that, for any 0 < ˇ < .0/ C  0 , the moment generating functions satisfy Q ."/ .ˇ; r/ < 1, r D 0; 1; : : :, for all " small enough. Indeed, condition C9 implies that Q ."/ ..0/ C  0 ; 0/ < 1 for " small enough, say .0/ 0 "  "1 . Let us denote cr D sups0 s r e . C ˇ /s < 1. Then, we have for "  "1 and r D 0; 1; : : : that Z 1 s r e ˇs FQ ."/ .ds/ Q ."/ .ˇ; r/ D 0

Z

 cr

1 0

e .

.0/ C 0 /s

FQ ."/ .ds/ D cr Q ."/ ..0/ C  0 ; r/ < 1:

Note that Q ."/ .; r/ are non-negative and non-decreasing functions of   0. Also, < .0/ C  , since ."/ ! .0/ as " ! 0. Thus, Q ."/ ..0/ ; r/, Q ."/ .."/ ; r/ < 1, r D 0; 1; : : :, for all " small enough, say "  "2 . Let us now formulate perturbation conditions for mixed power-exponential moment functionals for forcing functions: ."/

.k/ P17 : Q ."/ ..0/ ; r/ D Q .0/ ..0/ ; r/ C d1;r " C    C dkr;r "kr C o."kr / for r D 0; : : : ; k, where jdn;r j < 1, n D 1; : : : ; k  r, r D 0; : : : ; k.

It is convenient to define d0;r D Q .0/ ..0/ ; r/, r D 0; 1; : : : . .kC1/

.k/

.k/

Theorem 3.5.3. Let conditions D21 , C6 , F7 , P11 , P15 , D20 , C9 , and P17 hold. Then, for any A 2 0 , the following asymptotic expansions hold: QQ ."/ .A/ D QQ .0/ .A/ C h1 .A/" C    C hk .A/"k C o."k /;

(3.5.11)

where the coefficients hn .A/ are given by the formulas h0 .A/ D QQ .0/ ..0/ ; 0/ D f0000 f0 .A/, h1 .A/ D f0000 f1 .A/ C f1000 f0 .A/, and, for n D 0; : : : ; k, hn .A/ D

n X

fq000 fnq .A/

(3.5.12)

qD0

and fn000 D dn;0 C

n X

dnq;1 aq

qD1

C

X

n X

2mn qDm

dnq;m 

X

q1 Y

n1 ;:::;nq1 2Dm;q pD1

n

app =np Š:

(3.5.13)

160

3 Nonlinearly perturbed regenerative processes

Proof. We can write the asymptotic expansion of the first factor Q ."/ .."/ ; 0/ in the product expression for QQ ."/ .A/ given by (3.5.10) in powers of the difference ."/ D ."/  .0/ . Recall that ."/ ! 0. Hence, for any .0/ < ˇ < .0/ C  0 , the sum .0/ C j."/ j  ˇ for " small enough, say "  "3 . Condition C9 implies that, for any 0 < ˇ < .0/ C  0 , the moment generating functions satisfy Q ."/ .ˇ; r/ < 1; r D 0; 1; : : :, for " small enough, say "  "4 . By condition C9 , Z 1 Z 1 lim s kC1 e ˇs FQ ."/ .ds/  lim ckC1 e ıs FQ ."/ .ds/ < 1; (3.5.14) 0"!0 0

0"!0

0

kC1 .ıˇ /s

where ckC1 D sups0 s e < 1. It follows from relation .3:5:14/ that there exists "5 > 0 such that, for "  "5 , Z 1 1 Q sup s kC1 e sˇ FQ ."/ .ds/ < 1: (3.5.15) MkC1 D .k C 1/Š ""5 0 Let us once more use Taylor expansion (2.2.8) for the function e s , ."/ .0/ e s D e s 1 C s."/ =1Š C    C s k .."/ /k =kŠ

."/ ."/ C s kC1 .."/ /kC1 e sj j kC1 .s/=.k C 1/Š ;

(3.5.16)

."/ ."/ where kC1 .s/; s  0 is a continuous function such that 0  kC1 .s/  1, s  0. Define

"6 D min."3 ; "4 ; "5 /:

(3.5.17)

Substituting the Taylor expansion (3.5.16) into the integrals that define Q ."/ .."/ ; 0/, we obtain for "  "6 that Q ."/ .."/ ; 0/ D Q ."/ ..0/ ; 0/ C Q ."/ ..0/ ; 1/."/ C    C Q ."/ ..0/ ; k/.."/ /k =kŠ C .."/ /kC1 MQ kC1 QkC1 ; ."/

where

R1 ."/ QkC1 D

(3.5.18)

."/

s kC1 e s kC1 .s/FQ ."/ .ds/ 2 Œ0; 1: R1 sup""3 0 s kC1 e sˇ FQ ."/ .ds/ ."/

0

Let us use the asymptotic expansion ."/ D .0/ D a1 "C  Cak "k Co."k / given in Theorem 3.4.4. This expansion can be rewritten as an expansion for ."/ D ."/ .0/ . The functions Q ."/ ..0/ ; n/ can be represented in the form of asymptotic expansions, .k/ due to condition P17 . Substituting these expansions into (3.5.18) we obtain Q ."/ .."/ ; 0/ D .d0;0 C    C dk;0 "k C o."k // C .d0;1 C    C dk1;1 "k1 C o."k1 //  .a1 " C    C ak "k C o."k //=1Š C    C .d0;k C o.1//.a1 " C    C ak "k C o."k //k =kŠ C o."k /: (3.5.19)

3.5 Asymptotic expansions for quasi-stationary distributions

161

By grouping the coefficients at powers of " in this expansion, one can transform it to the forms described in (3.5.13), Q ."/ .."/ ; 0/ D f0000 C    C fk000 "k C o."k /:

(3.5.20)

The asymptotic expansion (3.5.6) for Q ."/ .A/ given in Theorem 3.5.2 has the following form: Q ."/ .A/ D f0 .A/ C f1 .A/" C    C fk .A/"k C o."k /:

(3.5.21)

Formulas and (3.5.20) and (3.5.21), in an obvious way, yield the asymptotic expansion (3.5.12) for the product QQ ."/ .A/ D Q ."/ .."/ ; 0/Q ."/ .A/. The corresponding coefficients can be found by grouping the coefficients of "n in the equation h0 C    C hk "k C o."k / D .f0000 C    C fk000 "k C o."k //.f0 .A/ C    C fk .A/"k C o."k //: (3.5.22) The proof is complete. As a corollary, we give asymptotic expansions for the quasi-stationary distribution  ."/ .A/ given by formula (3.3.11). Recall that  ."/ .A/ D

Q ."/ .A/ ; Q ."/ .X/

A 2 :

(3.5.23)

Note that Q ."/ .A/ D ! ."/ .."/ ; 0; A/= ."/ .."/ ; 1/ and, therefore, Z 1 Z 1 ."/ ."/ Q ."/ .X/ D e  s .1  FO ."/ .s// ds= se  s F ."/ .ds/: 0

0

Due to condition D21 , both distribution functions FO ."/ .s/ and F ."/ .s/ are not concentrated in zero for " small enough. Thus, Q ."/ .X/ ¤ 0 for such ". .kC1/

Theorem 3.5.4. Let conditions D21 , C6 , F7 , P11 A 2 0 , the following asymptotic expansions hold:  ."/ .A/ D

.k/

, and P15 hold. Then for any

Q .0/ .A/ C f1 .A/" C    C fk .A/"k C o."k / Q .0/ .X/ C f1 .X/" C    C fk .X/"k C o."k /

D  .0/ .A/ C g1 .A/" C    C gk .A/"k C o."k /;

(3.5.24)

where the coefficients gn .A/ are given by the recurrence formulas g0 .A/ D  .0/ .A/ D f0 .A/=f0 .X/, f0 .A/ D Q .0/ .A/, f0 .X/ D Q .0/ .X/ and, for n D 0; : : : ; k,   n1 X fnq .X/gq .A/ =f0 .X/: gn .A/ D fn .A/  qD0

(3.5.25)

162

3 Nonlinearly perturbed regenerative processes

Proof. In this case according formula (3.3.11),  ."/ .A/ D Q ."/ .A/=Q ."/ .X/. It follows from this formula that we can use the rational expansion (3.5.24) that can be transformed into a polynomial form. Note that Q .0/ .X/ ¤ 0. We have    k k X X .0/ r k .0/ r k gr .A/" C o." / Q .X/ C fr .X/" C o." /  .A/ C r D1

rD1

D Q .0/ .A/ C

k X

fr .A/"r C o."k /:

(3.5.26)

r D1

Relations (3.5.25) can be obtained by grouping the coefficients of "n in equation (3.5.26). Formulas (3.5.5) and (3.5.23) also give the following representation for the quasistationary distribution  ."/ .A/, alternative to (3.5.23):  ."/ .A/ D

! ."/ .."/ ; 0; A/ ; ! ."/ .."/ ; 0; X/

A 2 :

(3.5.27)

Representation (3.5.27) does not contain the functions  ."/ .."/ ; 1/. Using this, we can write an asymptotic expansion for quasi-stationary distributions, which is alterna.kC1/ tive to (3.5.24) and does not involve condition P11 . .k/

Theorem 3.5.5. Let conditions D21 , C6 , F7 , and P15 hold. Then, the following asymptotic expansions hold for any A 2 0 :  ."/ .A/ D

! .0/ ..0/ ; 0; A/ C f10 .A/" C    C fk0 .A/"k C o."k /

! .0/ ..0/ ; 0; X/ C f10 .X/" C    C fk0 .X/"k C o."k /

D  .0/ .A/ C g1 .A/" C    C gk .A/"k C o."k /;

(3.5.28)

where the coefficients fn0 .A/ are given by the formulas f00 .A/ D ! .0/ ..0/ ; 0; A/ D c0;0 .A/, f10 .A/ D c1;0 .A/ C c0;1 .A/a1 , and, for n D 0; : : : ; k, fn0 .A/ D cn;0 .A/ C

n X

cnq;1 .A/aq

qD1

C

X

n X

cnq;m .A/ 

2mn qDm

X

q1 Y

n

app =np Š ;

(3.5.29)

n1 ;:::;nq1 2Dm;q pD1

and the coefficients gn .A/ are given by the recurrence formulas g0 .A/ D  .0/ .A/ D f00 .A/=f00 .X/ and, for n D 0; : : : ; k,   n1 X 0 gn .A/ D fn0 .A/  fnq .X/gq .A/ =f00 .X/: (3.5.30) qD0

3.5 Asymptotic expansions for quasi-stationary distributions

163

Proof. In this case according formula (3.5.27),  ."/ .A/ D ! ."/ .."/ ; 0; A/=! ."/ .."/ ; 0; X/: This shows that we can write the rational expansion (3.5.28) that can be transformed to a polynomial form. Note that ! .0/ ..0/ ; 0; X/ ¤ 0. We have 

 .0/ .A/ C

k X

  k X gr .A/"r C o."k / ! .0/ ..0/ ; 0; X/ C fr .X/"r C o."k /

r D1

D ! .0/ ..0/ ; 0; A/ C

rD1 k X

fr0 .A/"r C o."k /:

(3.5.31)

r D1

Relations (3.5.30) can be obtained by grouping the coefficients of "n in equation (3.5.31). It is useful to note that despite the difference in the algorithms, the coefficients gn .A/, n D 1; : : : ; k, in both asymptotic expansions (3.5.24) and (3.5.28) coincide.

Chapter 4

Perturbed semi-Markov processes

Chapter 4 contains results concerning mixed ergodic theorems (for semi-Markov processes) and limit/large deviation theorems (for absorption times) for nonlinearly perturbed semi-Markov processes. We consider semi-Markov processes with a finite set of states X D f0; 1; : : : ; N g and the absorption state 0. The first hitting time to this state is the absorption time for the corresponding semi-Markov process. The asymptotic results mentioned above are given in the form of an asymptotics for joint distributions of positions of semi-Markov processes and absorption times. They are obtained by applying the corresponding results to regenerative processes given in Chapter 3. It can be done by using the fact that a semi-Markov process can be considered as a regenerative process with regeneration times which are subsequent return moments to any fixed state i ¤ 0. The first hitting time to the absorption state 0 is, in this case, the regenerative stopping time. We consider in details not only the generic case where the limiting semi-Markov process has one communication class of recurrent-without absorption states, but also the case, where the limiting semi-Markov process has one communication class of recurrent-without absorption states and, additionally, the class of non-recurrentwithout absorption states. The latter model covers a significant part of applications. We also formulate and prove a number of auxiliary propositions dealing with convergence of hitting probabilities, distributions, expectations, and exponential moments. We give propositions on so-called solidarity properties of cyclic absorption probabilities, and moment generating functions for hitting times for perturbed semi-Markov processes. These propositions, as we think, are useful by themselves. We consider a semi-Markov process ."/ .t /, t  0, with a phase space X D ."/ f0; 1; : : : ; N g and transition probabilities Qij .u/. The semi-Markov process ."/ .t /, t  0, is assumed to depend on a small perturbation parameter "  0. The processes ."/ .t /, t  0, for " > 0 are considered as perturbations of the process .0/ .t /, t  0, and therefore we assume some weak continuity conditions satisfied by transition quantities, namely, by the moment functionals on transition probabilities if regarded as as functions of " at the point " D 0. Our studies are concerned with the random functional ."/ 0 which is the first hitting time for the process ."/ .t /, t  0, to hit the absorption state 0. In applications, ."/ the hitting times 0 are often interpreted as transition times for different stochastic systems described by semi-Markov processes; these are occupation times or waiting

166

4 Perturbed semi-Markov processes

times in queuing systems, lifetimes in reliability models, extinction times in population dynamic models, etc. The object of our study is the joint distributions Pi f."/ .t / D j; ."/ 0 > t g and their asymptotic behaviour for t ! 1 and " ! 0. The mixed ergodic and limit theorems mentioned above describe the asymptotic behaviour of these joint distributions in the model with the one-step probability of absorption, f ."/ , which is averaged by the stationary distribution of the corresponding limiting embedded Markov chain, tending to 0 as " ! 0, and satisfying the condition that balances the rates for f ."/ ! 0 and time t ."/ ! 1 as " ! 0, f ."/ t ."/ !  as " ! 0;

(4.0.1)

while the corresponding mixed ergodic and limit theorems take the form of the following asymptotic relation for the generic case with one class of communication nonabsorption states for the limiting semi-Markov process: ."/

.0/

Pi f."/ .t ."/ / D j; 0 > t ."/ g ! j e =m

.0/

as " ! 0;

i; j ¤ 0;

(4.0.2)

where j.0/ , j ¤ 0, are stationary probabilities for the limiting semi-Markov process ."/ .t / and m.0/ is the mean value of the sojourn time averaged by the stationary distribution of the corresponding limiting embedded Markov chain. The asymptotic relation (4.0.2) means that the position in the semi-Markov process ."/ ."/ .t ."/ / and the normalised absorption time f ."/ 0 are asymptotically independent and have, in the limit, a stationary distribution and an exponential distribution, respectively. This can be interpreted as a mixed ergodic theorem (for semi-Markov processes) and a limit theorem (for absorption times). In the literature, the term transient is also used in connection with such a kind of asymptotic relations. For the asymptotic relation (4.0.2) to hold, we need to assume only the conditions of asymptotic ergodicity imposed on the semi-Markov processes ."/ .t /. These con."/ ditions, are based on the first moments for the distribution functions 0 Gjj .t / of the return-without-absorption times for some state j ¤ 0. A much stronger asymptotic relation can be obtained under an additional Cram´er type condition imposed on the exponential moments for distribution functions ."/ 0 Gjj .t /. In this case, the asymptotic relation (4.0.2) can be replaced with the following much stronger asymptotic relation: ."/ g Pi f."/ .t ."/ / D j; ."/ 0 >t ! QQ ij.0/ ..0/ / as " ! 0; ."/ ."/ expf t g

i; j ¤ 0;

where ."/ is given as a unique solution of the characteristic equation Z 1 ."/ ."/ e s 0 Gjj .ds/ D 1; 0

(4.0.3)

(4.0.4)

4

167

Perturbed semi-Markov processes .0/

and the limiting coefficients QQ ir ..0/ / are explicitly defined via moment generating ."/ functions for the distributions of the first hitting times (without absorption) 0 Gij .t / ."/

and 0 Gjj .t / at the point .0/ . There is a principal difference between the asymptotic relations (4.0.2) and (4.0.3). Firstly, there is no restriction on the rate with which t ."/ ! 1 as " ! 0 in the asymptotic relation (4.0.4). This relation allows for the case where f ."/ t ."/ ! 1, so that it can be interpreted as a mixed ergodic theorem (for semi-Markov processes) and a large deviation theorem (for absorption times). Secondly, the latter relation covers not only the case where (a) f ."/ ! f .0/ D 0 but also (b) f ."/ ! f .0/ > 0 as " ! 0. The first alternative (a) means that an absorption in 0 is impossible for the limiting process .0/ .t /, t  0, and it holds if and only if .0/ D 0. In such a case, the P

absorption times ."/ 0 ! 1 as " ! 0. The asymptotic relation (4.0.2) describes pseudo-stationary phenomena for perturbed semi-Markov processes. The second alternative (b) means that an absorption in 0 is possible for the limiting process .0/ .t /, t  0, and it holds if and only if .0/ > 0. In the latter case, the ."/ absorption times 0 are stochastically bounded as " ! 0. The asymptotic relation (4.0.3) describes quasi-stationary phenomena for perturbed semi-Markov processes. The sequential return times at any recurrent state i ¤ 0 are regeneration times for a semi-Markov process. The proofs of the asymptotic relations (4.0.2) and (4.0.3) is based on a use of this regeneration property for semi-Markov processes. The proofs mentioned above include, as parts, the so-called solidarity propositions. They show that the corresponding conditions for convergence formulated in terms of cyclic characteristics of return times, and the form of the corresponding asymptotic relations are invariant with respect to the choice of the recurrent state used to construct the regeneration cycles. In particular, we show that the characteristic root of ."/ does not depend on the choice of a recurrent state i ¤ 0. Another important part of these proofs clarifies the role of the initial state in the corresponding asymptotic relations. In the pseudo-stationary case (a), the limiting coefficients QQ .0/ ..0/ / in the asympij

totic relation (4.0.3) do not depend on the initial state i ¤ 0. Moreover, QQ ij.0/ ..0/ / D

j.0/ , i ¤ 0, i.e., these coefficients, as expected, coincide with the corresponding stationary probabilities of the limiting semi-Markov process .0/ .t /. The situation is not so simple in the quasi-stationary case (b). In this case, under some natural conditions, there exists a so-called quasi-stationary distribution for the semi-Markov process ."/ .t /, t  0, and it is given by the formula ."/

."/

Pi f."/ .t / D j=0 > t g ! j .."/ / as t ! 1;

i; j ¤ 0;

(4.0.5)

168

4 Perturbed semi-Markov processes

where ."/ j .."/ /

DP

.0/ .0/ QQ kj . /

QQ .0/ ..0/ / k¤0  kr

;

j ¤ 0:

In this case, the limiting coefficients QQ ij.0/ ..0/ / may depend on the initial state

i ¤ 0, but the quasi-stationary probabilities j."/ .."/ /, j ¤ 0, do not depend on the choice of the state k ¤ 0 in the formula above. We also give conditions for convergence of the quasi-stationary distributions of nonlinearly perturbed semi-Markov processes, ."/

.0/

j .."/ / ! j ..0/ / as " ! 0;

j ¤ 0:

(4.0.6)

Theorems 4.4.1 and 4.4.2 are mixed ergodic and limit theorems. Theorems 4.6.1 and 4.6.2 are mixed ergodic and large deviation theorems which cover the case of the pseudo-stationary asymptotics if the absorption probabilities are asymptotically small. Theorems 4.6.3, 4.6.5, 4.6.8, 4.6.7, and 4.6.11 cover the case of the so-called quasistationary asymptotics if the absorption probabilities can be asymptotically separated from zero. Theorems 4.4.5 and 4.6.6 give conditions for convergence of stationary and quasistationary distributions for perturbed semi-Markov processes. The mixed ergodic and large deviation theorems obtained in Chapter 4, together with the corresponding solidarity propositions for cyclic characteristics deduced for perturbed semi-Markov processes, create a basis for getting exponential expansions in the mixed ergodic and large deviation theorems. Such expansions are given in Chapter 5.

4.1 Semi-Markov processes with absorption In this section, we give a definition for semi-Markov processes and semi-Markov processes with absorption.

4.1.1 Semi-Markov processes Let .n ; n /, n D 0; 1; : : : be a homogeneous Markov chain, with a phase space X  Œ0; 1/, where X D f0; 1; : : : ; N g, an initial distribution Pf0 D i; 0 D 0g D pi , i 2 X, and transition probabilities PfnC1 D j; nC1  t =n D i; n D s; k D ik ; k D sk ; k D 0; : : : ; n  1g D PfnC1 D j; nC1  t =n D i; n D sg D Qij .t /;

i; j 2 X;

s; t 2 Œ0; 1/:

(4.1.1)

4.1 Semi-Markov processes with absorption

169

A characteristic property of the Markov chain .n ; n /, n D 0; 1; : : :, is that its transition probabilities do not depend on the current position of the second component. Such a Markov chain is called a Markov renewal process. In what follows, we use the notations Pi .A/ D PfA=0 D i g and Ei D Ef =0 D i g, respectively, to denote the conditional probabilities of events and expectations of the random variables determined by the Markov chain .n ; n /, n D 0; 1; : : : . Let us now define the random variables .0/ D 0 and .n/ D 1 C    C n for n  1, and then consider the stochastic processes .t / D .t/ ;

.t / D max.n W .n/  t /;

t  0:

A process .t /, t  0, defined in this way is called a semi-Markov process. The space X is referred to as a phase space of the semi-Markov process .t /. The probabilities Qij .t /, i; j 2 X, t  0, will be called transition probabilities of the semi-Markov process .t /, and the distribution pi , i 2 X, an initial distribution of this process. By the definition, the random variables n , n and n are, respectively, the successive moments of jumps, the times between the successive moments of jumps, and the positions of the process .t /, t  0, at successive moments of jumps. The random variable .t / counts the number of jumps of the process .t /, t  0, in the interval Œ0; t . As was mentioned, the transition probabilities of the Markov renewal process .n ; n / do not depend on a current position of the second component. This implies that the first component n , n D 0; 1; : : :, itself is a homogeneous Markov chain with the phase space X, the initial distribution Pf0 D i g D pi , i 2 X, and the transition probabilities PfnC1 D j=n D i g D pij D Qij .1/;

i; j 2 X:

(4.1.2)

This Markov chain is called an imbedded Markov chain for the semi-Markov process .t /, t  0. It is always possible to represent the transition probabilities Qij .t / in the following form: Qij .t / D pij Fij .t /;

i; j 2 X;

t  0;

(4.1.3)

where Fij .t / D Pf nC1  t =n D i; nC1 D j g;

i; j 2 X;

t 2 Œ0; 1/:

(4.1.4)

Note that the distribution function Fij .t / can be chosen in an arbitrary way if pij D 0. This choice does not affect the transition probability Qij .t /.

170

4 Perturbed semi-Markov processes

Let us also introduce the distribution functions of sojourn times, X X Qij .t / D pij Fij .t /; i 2 X: Fi .t / D j 2X

j 2X

We exclude from the consideration the case where the process .t /, t  0, has instant jumps by assuming the following condition: I1 : Fi .0/ D 0, i 2 X. Under this condition, the sequence of random variables .n/, n D 0; 1; : : :, is strictly increasing with probability 1. Moreover, it is easy to show that, in this case, P

.n/ ! 1 as n ! 1. This implies that .t / < 1 with probability 1 for every t  0, i.e., the semi-Markov process .t /, t  0, has a finite number of jumps in any finite interval.

4.1.2 Semi-Markov processes with absorption The following random variables play an important role in what follows. These random variables are j D minfn  1 W n D j g, which is the first hitting time for the imbedded Markov chain n in the state j , and j D .j /, which is the first hitting time for the semi-Markov process .t / in the state j . By the definition, the random variable j can take values 1; 2; : : : and 1, i.e., it may be a proper or improper integer-valued random variable. The random variable j can take values i the interval Œ0; 1, i.e., it may be a proper or improper integer-valued random variable. We assume that the sate 0 is an absorbing state if the following condition holds: A1 : p0j D 0, j ¤ 0. Under condition A1 , we have Pi f.t / D 0; t  0 =0 < 1g D 1, i 2 X. This explains the term absorption time applied to the random variable 0 . In this case it is natural to call .t /, t  0, a semi-Markov process with absorption. Let us introduce the hitting-without-absorption probabilities 0 fij

D Pi fj < 0 g;

i; j ¤ 0:

It is easy to show that 0 fij > 0 if and only if there exists a chain of states i D Qkij 1 j0 ; j1 ; : : : ; jkij D j such that j0 ; j1 ; : : : ; jkij ¤ 0 and nD0 pjn jnC1 > 0. We say that a state j ¤ 0 is a recurrent-without-absorption state if the hitting nonabsorption probability satisfies 0 fij > 0 for all i ¤ 0, i.e., the imbedded Markov chain n can reach, with positive probability, a state j from any state i before being absorbed in the state 0. Let us also introduce a class of recurrent-without-absorption states, X1 D fj ¤ 0 W

0 fij

> 0; i ¤ 0g:

4.1 Semi-Markov processes with absorption

171

We say that a state j ¤ 0 is a non-recurrent-without-absorption state if the hitting non-absorption probability satisfies 0 fij D 0 for some i ¤ 0, i.e., the imbedded Markov chain n can not reach with positive probability the state j from some state i ¤ 0 before being absorbed in the state 0. Let us also introduce a class of non-recurrent-without-absorption states, X2 D fj ¤ 0 W 0 fij D 0; for some i ¤ 0g: By the definition, X1 [ X2 D X0 D f1; : : : ; N g: Let also define absorption probabilities by jfi0

D Pi f0 < j g;

i; j ¤ 0:

The following conditions that guarantee existence of recurrent-without-absorption states will be used in the sequel: E1 : X1 ¤ ¿, i.e., there exists a state j ¤ 0 such that the hitting-without-absorption probability satisfies 0 fij > 0 for any state i ¤ 0. The most important is the case where the following condition holds: E01 : X1 D f1; : : : ; N g; X2 D ¿, i.e., 0 fij > 0 for all i; j ¤ 0. However, we will also consider an intermediate case where the following condition holds: E001 : X1 ; X2 ¤ ¿, i.e., there exist j; r ¤ 0 such that 0 fij > 0 for every i ¤ 0, and 0 fir D 0 for some i ¤ 0.

4.1.3 Classification of states Let us now classify communicative properties of a semi-Markov process .t /, t  0, and the corresponding imbedded Markov chain n , n D 0; 1; : : :, in traditional terms (see, for example, Feller (1950, 1968)). We introduce non-conditional hitting probabilities by fij D Pi fj < 1g;

i; j 2 X:

By the definition, the hitting probability satisfies fij > 0 if and only if there exists Qkij 1 a chain of states i D j0 ; j1 ; : : : ; jkij D j such that nD0 pjn jnC1 > 0. The following four cases have to be considered: (i) Condition E01 holds and pi0 > 0 for some state i 2 X1 . In this case, X1 is a non-closed communicative class of non-essential states, while f0g is a closed communicative class with one essential state 0. The hitting probabilities are: (a) fij > 0, i; j 2 X1 , and fi0 > 0, i 2 X1 ; (b) f00 D 1; f0j D 0, j 2 X1 .

172

4 Perturbed semi-Markov processes

(ii) Condition E01 holds and pi0 D 0 for every state i 2 X1 . In this case, X1 and f0g are two closed communicative classes of essential states. The hitting probabilities are: (c) fij > 0, i; j 2 X1 , while fi0 D 0, i 2 X1 ; (d) f00 D 1; f0j D 0, j 2 X1 . (iii) Condition E001 holds and pi0 > 0 for some state i 2 X1 . In this case, X1 is a non-closed communicative class of non-essential states, X2 is a non-closed class of non-essential states, while f0g is a closed communicative class with one essential state 0. The hitting probabilities are: (e) fij > 0, i; j 2 X1 and fi0 > 0, i 2 X1 , while fij D 0, i 2 X1 , j 2 X2 ; (f) fij > 0, i 2 X2 , j 2 X1 , while fi0 2 Œ0; 1/, i 2 X2 ; (g) f00 D 1; f0j D 0, j 2 X1 [ X2 . (iv) Condition E001 holds and pi0 D 0 for every state i 2 X1 . In this case, X1 and f0g are two closed communicative classes of essential states, while X2 is a non-closed class of non-essential states. The hitting probabilities are: (h) fij > 0, i; j 2 X1 , while fi0 D 0, i 2 X1 and fij D 0, i 2 X1 , j 2 X2 ; (i) fij > 0, i 2 X2 , j 2 X1 , while fi0 2 Œ0; 1/, i 2 X2 ; (j) f00 D 1; f0j D 0, j 2 X1 [ X2 . Note that alternatives (i) and (iii) correspond to a model in which absorption is possible from recurrent states, while alternatives (ii) ad (iv) correspond to a model in which absorption from recurrent states is not possible. Let us consider the random variables j ^ 0 , j ¤ 0. By the definition of these random variables, Pi fj ^ 0 < 1g D 0 fij C jfi0 ;

i; j ¤ 0:

(4.1.5)

The following useful lemma follows directly from the classification of the states described above. Lemma 4.1.1. If j ¤ 0 is a recurrent-without-absorption state, then the random variable i ^ 0 is finite with probability 1, i.e., 0 fij

C jfi0 D 1:

(4.1.6)

4.1.4 Removing the absorption Let F t D ..s/, s 2 Œ0; t /, t  0, be a natural filtration of the -algebras of the random events generated by the semi-Markov process .t /. Suppose that we are interested in finding the probabilities Pi fA t ; 0 > t g, t  0, for some random events A t 2 F t , t  0, and an initial state i ¤ 0. It is easy to see that this probability does not depend on the transition probabilities Q0k .u/, u  0, k 2 X. So, we can change these transition probabilities without changing values of the probabilities Pi fA t ; 0 > t g, t  0. In particular, this remarks can be applied to the probabilities Pi f.t / D j; 0 > t g, t  0, where i; j ¤ 0. These probabilities make a subject of our studies in Chapter 4. One of natural ways to change transition probabilities is for the absorption state to take new values, Q0k .u/ D N 1C1 .1  e u /, u  0, k 2 X. For the new modified version of the semi-Markov process .t /, the state 0 is not an absorbing state any more.

4.2 Perturbed semi-Markov processes

173

Note that this change of transition probabilities in the state 0 also changes the classification of states described in Subsection 4.1.3. In the cases (i) and (iii), the whole phase space X D X1 [ f0g becomes a single communicative class of essential states. In the case (ii), X1 becomes a closed communicative class of essential states, while the state 0 becomes non-essential. In case (iv), X1 becomes a closed communicative class of essential states, while the class X2 [ f0g becomes a non-closed class of non-essential states.

4.1.5 Instant transitions Condition I1 , which prohibits existence of instant states, can be weakened in the case where condition E1 holds. Condition I1 can be replaced with any condition that would imply Pi .A/ D 1, i 2 X, where the random event is A D flimn!1 .n/ D 1g. Indeed, in this case the random variable satisfies Pi f.t / < 1; t  0g D 1, i 2 X, and, therefore, the semi-Markov process .t / D .t/ , t  0, will be well defined for t  0. Lemma 4.1.1 implies that, under condition E1 , condition I1 can be replaced with the following much weaker condition that allows for instant transitions: I2 : (a) Fi .0/ < 1 for some state i 2 X1 ; (b) F0 .0/ < 1. Let us define return times by r .n/ D min.n > r .n  1/ W n D r/, n D 1; 2; : : : ; r .0/ D 0. By the definition, j .n/ is the time of n-th return into the state r for the imbedded Markov chain n . Let also condition E1 hold and j 2 X1 . If 0 D 0, then n , n D 1; 2; : : :, are i.i.d. random variables with the distribution function F0 .t /. Thus, condition I2 (b) implies that P0 .A/ D 1. In the cases (i) and (iii), the random variable satisfies fi0 D Pi f0 .1/ < 1g D 1, i 2 X. Therefore, Pi .A/ D fi0 P0 .A/ D 1 for any i 2 X. In the cases (ii) and (vi), if j 2 X1 , then Pj fj .n/ < 1, n D 1; 2; : : :g D 1, and the random variables j .n/C1 , n D 1; 2; : : :, are i.i.d. random variables with the distribution function Fj .t /. Thus, condition I2 (a) implies that Pj .A/ D 1. In this case, using Lemma 4.1.1, we get Pi .A/ D jfi0 P0 .A/C 0 fij Pj .A/ D jfi0 C 0 fij D 1 for any i 2 X.

4.2 Perturbed semi-Markov processes In this section, we consider a model of perturbed semi-Markov processes, and study convergence of some basic characteristics of return times for such processes.

4.2.1 Perturbed semi-Markov processes ."/

."/

Let, for every "  0, .n ; n /, n D 0; 1; : : : be a Markov renewal process with the ."/ phase space X D f0; 1; : : : ; N g and transition probabilities Qij .u/.

174

4 Perturbed semi-Markov processes ."/

."/

Let 0 D ."/ .0/; ."/ .n/ D 1 C    C n , n  1, and  ."/ .t / D max.n W ."/

."/ .n/  t /, t  0, and ."/ .t / D  ."/ .t/ , t  0, be the corresponding semi-Markov ."/

."/

process associated with the Markov renewal process .n ; n /. ."/ ."/ ."/ ."/ We write the transition probabilities as Qij .t / D pij Fij .t /, where pij D ."/ .1/ are the transition probabilities of the corresponding imbedded Markov chain Qij

and Fij."/ .t / are the distribution functions of transition times.

."/ Let also j."/ D minfn  1 W ."/ D ."/ .j."/ / be the first hitting n D j g and j

."/ time at which the imbedded Markov chain ."/ n and the semi-Markov process  .t / ."/ hit the state j , respectively. The first hitting time 0 is the absorption time for the semi-Markov process ."/ .t /, t  0. ."/ ."/ ."/ It is also useful to define return times by j .n/ D min.n > j .n1/ W n D j /, ."/

."/

."/

n D 1; 2; : : : ; j .0/ D 0 and j .n/ D ."/.j .n//, n D 0; 1; : : : . By the defi."/

."/

nition, j .n/ and j .n/ are the times of the n-th return into the state j for the ."/

imbedded Markov chain n and the semi-Markov process ."/ .t /, respectively. Denote X ."/ X ."/ ."/ ."/ Qij .t / D pij Fij .t /; i 2 X: Fi .t / D j 2X

j 2X

For simplicity we assume that the following condition preventing instant jumps holds: ."/

I3 : Fi .0/ D 0, i 2 X, for all "  0. We consider a model in which 0 is an absorbing state, that is, the following condition holds: ."/

A2 : p0j D 0, j ¤ 0, for all "  0. We also assume the following condition imposed on the transition probabilities of the semi-Markov process ."/ .t /, t  0, which allows to consider these processes for " > 0 as perturbed versions of the semi-Markov process .0/ .t /, t  0, T1 :

."/

.0/

(a) pij ! pij as " ! 0, i ¤ 0, j 2 X; ."/ .0/ ./ ) Qij ./ as " ! 0, i ¤ 0, j 2 X. (b) Qij

The following condition obviously implies condition T1 to hold: T01 :

."/

.0/

(a) pij ! pij as " ! 0, i ¤ 0, j 2 X; ."/

.0/

(b) Fij ./ ) Fij ./ as " ! 0, i ¤ 0, j 2 X.

175

4.2 Perturbed semi-Markov processes

It is useful to note that the asymptotic relations in conditions T1 and T01 are equiv.0/ alent for i ¤ 0, j 2 X such that pij > 0. However, it can be that the asymptotic relation (b) in condition T1 holds while the corresponding asymptotic relation (b) in .0/ ."/ .0/ condition T01 does not hold, if pij D 0. It is so, if pij ! pij D 0 as " ! 0 but

."/ .0/ ./ ) Qij ./ as " ! 0, where Fij."/ ./ 6) Fij.0/ ./ as " ! 0. Indeed, in this case, Qij .0/ .t / D 0, t  0. Qij We also assume the following recurrence condition for the limit semi-Markov process:

E2 : The set of recurrent-without-absorption states for the semi-Markov process .0/ .t / is not empty, X1.0/ ¤ ¿, i.e. there exists a state j ¤ 0 such that the hitting-without-absorption probability satisfies 0 fij.0/ > 0 for any state i ¤ 0. The most important is the case where the following condition holds: .0/

E02 : X1

.0/

D f1; : : : ; N g; X2

.0/

D ¿, i.e., 0 fij

> 0 for all i; j ¤ 0.

However, we also consider an intermediate case where the following condition holds: .0/

.0/

E002 : X1 ; X2 and

.0/ 0 fir

.0/

¤ ¿, i.e., there exist j; r ¤ 0 such that 0 fij

> 0 for every i ¤ 0,

D 0 for some i ¤ 0.

4.2.2 Reduction to the model of perturbed regenerative processes We are interested in an asymptotic analysis for the probabilities ."/

."/

Pij .t / D Pi f."/ .t / D j; 0 > t g;

t  0;

i; j ¤ 0:

These probabilities are measurable functions of t  0 with respect to the Borel -algebra of subsets of Œ0; 1/ for every i; j ¤ 0. The semi-Markov process ."/ .t /, t  0, can be considered as a particular case of ."/ a regenerative process with regeneration times j .n/, which are times of subsequent return of the process ."/ .t / into some fixed recurrent state j . ."/ The absorption time 0 plays a role of the regenerative stopping time. However, we have to be careful. Under the absorption condition A2 , the process ."/ .t / is a regenerative process with improper regeneration times that can take the value C1 (in the case of absorption). In Chapter 3, we have considered regenerative processes with proper regeneration times. Thus, the procedure of removing the absorption, described in Subsection 4.1.4, should be applied first. For example, transition probabilities in the absorption state can be changed and ."/ take new values such as Q0k .u/ D N 1C1 .1  e u /, u  0, k 2 X. For the new modified version of the semi-Markov process ."/ .t /, the state 0 is not absorbing any

176

4 Perturbed semi-Markov processes

more. As was mentioned in Subsection 4.1.3, the initial and the modified semi-Markov processes have the same probabilities, Pij."/ .t / D Pi f."/ .t / D j; ."/ 0 > t g, t  0, i; j ¤ 0. As we will show below, conditions A2 , T1 and E2 imply that condition E2 holds for the semi-Markov process ."/ .t /, t  0, moreover, all the states j 2 X1 are recurrentwithout-absorption states for this process, if " is small enough. If condition E2 holds for the semi-Markov process ."/ .t /, t  0, then there exists a unique closed communicative class of essential states for the corresponding modified version of this semi-Markov process, and any recurrent state j 2 X1 belongs to this class. ."/ This implies that the return times j .n/, n D 0; 1; : : :, are, with probability 1, finite random variables for the modified version of the semi-Markov process ."/ .t /. Let us fix some state j 2 X1.0/ . If the initial state is i D j , then the process ."/ .t /, t  0, is a standard regenerative process. However, if the initial state is i ¤ j , the process ."/ .t /, t  0, is a regenerative process with a transition period. Let us introduce the distribution functions for the first hitting times, ."/ 0 Gij .t /

."/

D Pi fj

."/

 t; j

."/

 0 g;

t  0;

i; j ¤ 0;

and the distribution functions for the absorption times, ."/ j Gi0 .t /

D Pi fj."/  t; 0."/  j."/ g;

t  0;

i; j ¤ 0:

Using the regeneration property, which the semi-Markov process ."/ .t /, t  0, possesses at the time of the first hitting into any state, we can write the following renewal equation for the case where the initial state is i D j ¤ 0: Z t ."/ Pjj."/ .t / D 1  Fj."/ .t / C Pjj."/ .t  s/0 Gjj .ds/; t  0: (4.2.1) 0

It is worth to note a very simple form the forcing function 1  Fj."/ .t / takes in

equation (4.2.1). As a matter of fact, if the initial state is j , then f."/ .t / D j; j."/ ^

."/ .1/ > t g, that is, the process ."/ .t / can be observed in the state j at ."/ 0 > t g D f

the moment t before the time it hits this state for the first time or being absorbed, only if no jumps occur before the moment t . As was mentioned above, the semi-Markov process ."/ .t /, t  0, considered as a regenerative process, has a transition period in the case where the initial state is i ¤ j; 0. The renewal equation (4.2.1) should be supplemented, in this case, with the following renewal type relation that can be written for every initial state i ¤ j , i; j ¤ 0: Z t ."/ ."/ ."/ Pjj .t  s/ 0 Gij .ds/; t  0: (4.2.2) Pij .t / D 0

177

4.2 Perturbed semi-Markov processes

Note that there is no free forcing function in this renewal type relation. As a matter of fact, if the initial state is i ¤ j; 0, then f."/ .t / D j; j."/ ^ ."/ 0 > t g D ¿. The renewal relations (4.2.1) and (4.2.2) can be used to make an asymptotic analysis ."/ ."/ of the probabilities Pij .t / in the case where j 2 X1 , i.e., it is a recurrent-withoutabsorption state. ."/ In a general case, for any state r ¤ 0 and j 2 X1 , the following renewal equation can be used: Z t ."/ ."/ ."/ ."/ Pjr .t / D qjr .t / C Pjr .t  s/0 Gjj .ds/; t  0; (4.2.3) 0

where

."/

."/

qjr .t / D Pj f."/ .t / D r; j

."/

> t; 0 > t g;

t  0:

It is obvious that ."/ .t / D Pi fj."/ ^ ."/ Gij."/ .t / D 0 Gij."/ .t / C j Gi0 0  t g; t  0; i; j ¤ 0;

(4.2.4)

and that the following estimate takes place: ."/ qjr .t /  1  Gjj .t /;

t  0;

j; r ¤ 0:

(4.2.5)

The renewal equation (4.2.3) should be supplemented with the following renewal type relation that can be written for every initial state i ¤ j; 0 as Z t ."/ ."/ ."/ ."/ Pjr .t  s/0 Gij .ds/; t  0; (4.2.6) Pir .t / D qijr .t / C 0

where

."/

."/

qijr .t / D Pi f."/ .t / D r; j

."/

> t; 0 > t g;

t  0:

It is obvious that the following estimate takes place: ."/

qijr .t /  1  Gij .t /;

t  0;

i; j; r ¤ 0:

(4.2.7)

The renewal relations (4.2.1) and (4.2.2) are simpler than the renewal relations (4.2.3) and (4.2.6), because they have simpler forcing functions. The first two re."/ newal relations can be used in the asymptotic analysis of the probabilities Pij .t / in ."/

the case where j 2 X1 , i.e., j is a recurrent-without-absorption state for the Markov chain ."/ n . The latter two renewal relations can be used, in any case, for an asymptotic ."/ analysis of the probabilities Pir .t /. However, it makes sense to use them in the case where X2."/ ¤ ¿ and r 2 X2."/ . We can now apply results of Chapters 1–3 to these equations. To do this, we should translate the corresponding perturbation conditions imposed on the transition probabil."/ ."/ ities Qij .t / of perturbed semi-Markov processes to the distribution functions 0 Gij .t / ."/

and the forcing functions ı.i; j /.1  Fj .t // used in (4.2.1) and (4.2.2).

178

4 Perturbed semi-Markov processes

It should be noted that the renewal equation (4.2.1) and the renewal relations (4.2.2) are the same for the modified and the initial versions of the semi-Markov process ."/ .t /, and all asymptotic results can be related to the initial version of the semiMarkov process ."/ .t /.

4.2.3 Convergence of hitting and absorption probabilities Let us introduce the hitting and absorption probabilities by

and

."/ 0 fij

D Pi fj

."/

 0 g D 0 Gij .1/;

."/

."/

i; j ¤ 0;

."/ jfi0

."/ D Pi f0."/  j."/ g D j Gi0 .1/;

i; j ¤ 0:

Lemma 4.2.1. Let conditions A2 , T1 (a), and E2 hold. Then, .˛/ 0 fij."/ ! 0 fij.0/ as " ! 0, i ¤ 0, j 2 X1.0/ ; .ˇ/ jfi0."/ ! jfi0.0/ as " ! 0, i ¤ 0, j 2 X1.0/ . ."/

Proof. The probabilities 0 fij , i ¤ 0, satisfy the following system of linear equations for every j ¤ 0: X ."/ ."/ ."/ ."/ pik 0 fkj ; i ¤ 0: (4.2.8) 0 fij D pij C k¤j;0

Systems (4.2.8) can be rewritten, for every j ¤ 0, in the following matrix form: ."/ 0 fj

where

2 ."/ 0 fj

and

6 D6 4

."/ ."/ D p."/ j C j P 0 fj ;

."/ 0 f1j

2

3

:: 7 : 7 5; ."/ 0 fNj

(4.2.9)

."/

pj

3 ."/ p1j 6 : 7 : 7 D6 4 : 5; ."/ pNj

2

jP

."/

3 ."/ ."/ ."/ ."/ p11 : : : p1.j 1/ 0 p1.j C1/ : : : p1N 6 : :: :: :: :: 7 7 : D6 : : : : 5: 4 : ."/ ."/ ."/ ."/ pN1 : : : pN.j 1/ 0 pN.j C1/ : : : pN N

Let us show that the coefficient matrix of this system, I  j P."/ , for every j 2 X1 has an inverse for all " small enough, and, therefore, a unique solution of the systems (4.2.9) has the following form: ."/ 0 fj

D ŒI  j P."/ 1 p."/ j :

(4.2.10)

179

4.2 Perturbed semi-Markov processes

We introduce the probabilities ."/ 0jfi .n/

."/

."/

D Pi f0 ^ j

> ng;

i; j ¤ 0;

n D 0; 1; : : : :

Condition T1 (a) implies that, for i; j ¤ 0, n D 0; 1; : : :, ."/ 0jfi .n/

n Y

X

D

iDi0 ;:::;in ¤0;j kD1

!

.0/ 0jfi .n/

."/

pik1 ik X

D

n Y

iDi0 ;:::;in ¤0;j kD1

.0/

pik1 ik as " ! 0:

(4.2.11)

Condition E2 implies, by Lemma 4.1.1, that for j 2 X1 , .0/ 0jfi .n/

! 0 as n ! 0;

i ¤ 0;

(4.2.12)

and thus there exists m such that .0/

max 0jfi i¤0

(4.2.13)

.m/ < L < 1:

It follows from (4.2.11) and (4.2.13) that there exists "0 > 0 such that sup max

""0 i¤0

."/ 0jfi .m/

 L:

(4.2.14)

Obviously, ."/ 0jfi .2m/

D

X

."/ Pi fj."/ ^ 0."/ > m; ."/ m D kg 0jfk .m/:

(4.2.15)

k¤0;j

Using estimate (4.2.14) and relation (4.2.15) we get sup max

""0 i¤0

."/ 0jfi .nm/

< Ln ;

n D 1; 2; : : : :

(4.2.16)

Now introduce the probabilities ."/ 0jfik .n/

."/

D Pi f."/ n D k; j

."/

^ 0 > ng;

i; k; j ¤ 0;

n D 0; 1; : : : :

It is easily seen that the n-th power of the matrix j P."/ takes the following form: ."/

.j P."/ /n D k0jfik .n/k ;

n D 0; 1; : : : :

(4.2.17)

180

4 Perturbed semi-Markov processes

For every j 2 X1 , estimate (4.2.16) yields the following estimate for the norm of the matrix .j P."/ /n , and the estimate is uniform for "  "0 : j .j P."/ /n j D max i¤0

D max i¤0

N X

."/ 0jfik .n/

kD1 ."/ 0jfi .n/

 LŒn=m ! 0 as n ! 0:

(4.2.18)

Relation (4.2.18) implies that for every j 2 X1 and every "  "0 there exists an inverse matrix, ŒI  j P."/ 1 , given by the following matrix series, which is norm convergent: ŒI  j P."/ 1 D I C j P."/ C .j P."/ /2 C    :

(4.2.19)

."/

For every j; k ¤ 0, let us introduce a random variable ıj k as the number of visits ."/

of the imbedded Markov chain n in the state k before the first hitting of one of the sates j or 0, ."/ ıj k

j."/ ^0."/

X

D

."/

.n1 D k/:

nD1

By the definition, ."/

Ei ıj k D

1 X

."/

."/

Pi fn1 D k; j

."/

^ 0 > n  1g:

(4.2.20)

nD1 ."/

Finally, comparing formulas (4.2.19) and (4.2.20) we get that Ei ıj k < 1, i; k ¤ 0, .0/

.0/

j 2 X1 , and that for every j 2 X1

and "  "0 , ."/

ŒI  j P."/ 1 D k Ei ıj k k:

(4.2.21)

Condition T1 implies in an obvious way that, for every n D 0; 1; : : :, .j P."/ /n ! .j P.0/ /n as " ! 0:

(4.2.22)

Relations (4.2.18) and (4.2.22) imply that ŒI  j P."/ 1 ! ŒI  j P.0/ 1 as " ! 0:

(4.2.23)

One can prove in the following simple way that the inverse matrix ŒI  j P."/ 1 exists for all " small enough and that convergence relation (4.2.25) holds true. As was mentioned above, the inverse matrix ŒI  j P.0/ 1 does exist. This means that det.I  j P.0/ / ¤ 0. Condition T1 obviously implies that j P."/ ! j P.0/ as " ! 0. Thus,

4.2 Perturbed semi-Markov processes

181

det.I  j P."/ / ¤ 0 for all " small enough. Moreover, the elements of the inverse matrix ŒI  j P."/ 1 are continuous rational functions of elements of the matrix I  j P."/ . This rational function has the non-zero denominator det.I  j P."/ /. This implies existence of ŒI  j P."/ 1 for " small enough and relation (4.2.25). .0/ Formulas (4.2.10) and (4.2.21) imply that, for every j 2 X1 and "  "0 , ."/ 0 fij

N X

D

Ei ıj."/ p ."/ ; k kj

i ¤ 0:

(4.2.24)

i; j; k ¤ 0:

(4.2.25)

kD1

According to relation (4.2.23), ."/

.0/

Ei ıj k ! Ei ıj k as " ! 0;

Relation (4.2.25), condition T1 (a) and formula (4.2.24) imply the convergence relation .˛/ in Lemma 4.2.1. For every j ¤ 0, the probabilities jfi0."/ , i ¤ 0, satisfy the following system of linear equations that are similar to (4.2.8): X ."/ ."/ ."/ ."/ pik jfk0 ; i ¤ 0: (4.2.26) jfi0 D pi0 C k¤j;0

System (4.2.26) can be rewritten, for every j ¤ 0, in the following matrix form: ."/ j f0

where

2 ."/ j f0

6 D4

."/

."/

D p0 C j P."/ j f0 ;

."/ jf10

:: :

3 7 5;

(4.2.27)

."/ 3 p10 7 6 D 4 ::: 5 :

2

p."/ 0

."/ jfN 0

."/

pN 0

This system has the same coefficient matrix I  j P."/ as system (4.2.9) for every .0/ j 2 X1 . As was shown in the proof of Lemma 4.2.10, this matrix has an inverse for "  "0 . Therefore, a unique solution of the system (4.2.27) has the following form: ."/ j f0

D ŒI  j P."/ 1 p."/ 0 :

(4.2.28)

Taking into account formula (4.2.21) and (4.2.28) we get the following formula for .0/ every j 2 X1 and "  "0 as above: ."/ jfi0

D

N X

Ei ıj."/ p ."/ ; k k0

i ¤ 0:

(4.2.29)

kD1

Relation (4.2.25), condition T1 (a), and formula (4.2.29) imply the convergence relation .ˇ/ in Lemma 4.2.1.

182

4 Perturbed semi-Markov processes

The following lemma is a direct corollary of Lemma 4.2.1. Lemma 4.2.2. Let conditions A2 , T1 (a), and E2 hold. Then, any state j 2 X1.0/ is also a recurrent-without-absorption state for the perturbed semi-Markov process ."/ .t /, t  0, for all "  "0 (defined in formula (4.2.14)), i.e., X1.0/ X1."/ for all "  "0 .

4.2.4 Weak convergence of distribution functions of the first hitting times The following lemma gives a condition for weak convergence of the distribution functions of the first hitting times. ."/

.0/

Lemma 4.2.3. Let conditions A2 , T1 , and E2 hold. Then .˛/ 0 Gij ./ ) 0 Gij ./ as .0/

."/

.0/

.0/

" ! 0, i ¤ 0, j 2 X1 ; .ˇ/ j Gi0 ./ ) j Gi0 ./ as " ! 0, i ¤ 0, j 2 X1 ; ."/ .0/ .0/ ./ Gij ./ ) Gij ./ as " ! 0, i ¤ 0, j 2 X1 . Proof. Let us use the Laplace transform, Z 1 Z ."/ ."/ s ."/ N pNij ./ D e Qij .ds/; 0 ij ./ D 0

1 0

."/

e s 0 Gij .ds/;   0; i; j ¤ 0:

Note that condition T1 can be expressed in the following equivalent form: ."/

.0/

T001 : pNij ./ ! pNij ./ as " ! 0 for   0, i ¤ 0, j 2 X. The quantities 0 Nij."/ ./, i ¤ 0, for every j ¤ 0 and   0, satisfy the following system of linear equations: X ."/ ."/ ."/ ."/ pNik ./ 0 N kj ./; i ¤ 0: (4.2.30) 0 N ij ./ D pNij ./ C k¤j;0

System (4.2.30), for every j ¤ 0, can be rewritten in the following matrix form: N ."/ 0 ¥j ./ where

and

2

N ."/ N ."/ D pN ."/ j ./ C j P ./ 0 ¥j ./; 2

3 ."/ ./ pN1j 6 6 7 7 :: :: 6 7 ; pN ."/ ./ D 6 7; N ."/ 0 ¥j ./ D 4 : : j 4 5 5 ."/ ."/ N pNNj ./ 0 Nj ./ 2

N ."/ ./ 0 1j

3

(4.2.31)

3 ."/ ."/ ."/ ."/ pN11 ./ : : : pN1.j 1/ ./ 0 pN1.j C1/ ./ : : : pN1N ./ 6 7 :: :: :: :: :: 7: N ."/ ./ D 6 jP : : : : : 4 5 ."/ ."/ ."/ ."/ ./ : : : pNN.j ./ 0 p N ./ : : : p N ./ pNN1 NN 1/ N.j C1/

183

4.2 Perturbed semi-Markov processes

Let us show that the coefficient matrix of this system, I  j PN ."/ ./, for every j 2 X1.0/ and   0, has an inverse for all " small enough, and, therefore, a unique solution of system (4.2.31) has the following form: N ."/ 0 ¥j ./

."/ D ŒI  j PN ."/ ./1 pN j ./:

(4.2.32)

Indeed, by the definition, the elements of the matrices j P."/ ./ and j P."/ satisfy the following inequalities for every   0: ."/ ."/ 0  .k ¤ j /pNik ./  .k ¤ j /pik ;

i; k ¤ 0;

(4.2.33) .0/

and thus the corresponding norms satisfy the following relation for every j 2 X1 ,   0 and "  "0 : j .j PN ."/ .//n j  j .j PN ."/ /n j  LŒn=m ! 0 as n ! 0:

(4.2.34)

.0/

Relation (4.2.34) implies that for every j 2 X1 ;   0, and every "  "0 there exists an inverse matrix, ŒI  j P."/ ./1 , given by the following (convergent in the norm) matrix series, ŒI  j PN ."/ ./1 D I C j PN ."/ ./ C .j PN ."/ .//2 C    :

(4.2.35)

For every j; k ¤ 0,   0, let us introduce random variables by ."/ ıNj k ./ D

1 X

e 

."/ .n1/

."/

."/

.n1 D k; j

."/

^ 0 > n  1/:

nD1

By the Lebesgue theorem, Ei ıNj."/ ./ D k

1 X

Ei e 

."/ .n1/

."/ ."/ .."/ n1 D k; j ^ 0 > n  1g:

(4.2.36)

nD1

Finally, by comparing formulas (4.2.35) and (4.2.36), we get that Ei ıNj k ./ < 1, ."/

i; k ¤ 0, j 2 X1.0/ ,   0, and that for every j 2 X1.0/ ,   0, and "  "0 , ./ k: ŒI  j PN ."/ ./1 D k Ei ıNj."/ k

(4.2.37)

Condition T1 implies in an obvious way that, for every j 2 X1 ,   0, N ."/ j P ./

! j PN .0/ ./ as " ! 0:

(4.2.38)

As was mentioned above, the inverse matrix ŒI  j PN ."/ ./1 exists for all " small enough. Moreover, elements of the inverse matrix ŒI  j PN ."/ ./1 are continuous

184

4 Perturbed semi-Markov processes

rational functions of elements of the matrix I  j PN ."/ ./. This rational function has the non-zero denominator det.I  j PN ."/ .//. This implies that, for every j 2 X1.0/ ,   0, ŒI  j PN ."/ ./1 ! ŒI  j PN .0/ ./1 as " ! 0:

(4.2.39)

.0/

Formulas (4.2.32) and (4.2.37) imply that, for every j 2 X1 ,   0, and "  "0 , N ."/ ./ 0 ij

D

N X

."/ ."/ Ei ıNj k ./pNkj ./;

i ¤ 0:

(4.2.40)

i; j; k ¤ 0:

(4.2.41)

kD1

According to relation (4.2.39), for every   0, Ei ıNj k ./ ! Ei ıNj k ./ as " ! 0; ."/

.0/

.0/

Relation (4.2.41), condition T001 , and formula (4.2.40) imply that, for every j 2 X1 and   0, N ."/ ./ 0 ij

! 0 Nij."/ ./ as " ! 0;

i ¤ 0:

(4.2.42)

This relation is equivalent to the weak convergence relation .˛/ in Lemma 4.2.3. The proof of the weak convergence relation .ˇ/ given in Lemma 4.2.3 repeats in main the proof given above. Let us now use the Laplace transforms Z 1 Z 1 ."/ ."/ ."/ ."/ pNi0 ./ D e s Qi0 .ds/; j N i0 ./ D e sj Gi0 .ds/;   0; i; j ¤ 0: 0

0

."/

For every j ¤ 0 and   0, the quantities j N i0 ./, i ¤ 0, satisfy the following system of linear equations: X ."/ ."/ ."/ ."/ pNik ./ j Nk0 ./; i ¤ 0: (4.2.43) j N i0 ./ D pNi0 ./ C k¤j;0

For every j ¤ 0 and   0, system (4.2.43) can be rewritten in the following matrix form: N ."/ j ¥0 ./ where

2 N ."/ j ¥0 ./

6 D4

N ."/ N ."/ D pN ."/ 0 ./ C j P ./j ¥0 ./;

N ."/ j 10 ./ :: :

N ."/ j N 0 ./

3 7 5;

(4.2.44)

3 ."/ pN10 ./ 7 6 ."/ :: pN 0 ./ D 4 5: : 2

."/

pNN 0 ./

For every j 2 X1 and   0, this system has the same coefficient matrix I  j PN ."/ ./ as system (4.2.31). As was shown in the proof of Lemma 4.2.10, this matrix has an

4.2 Perturbed semi-Markov processes

185

inverse for "  "0 . Therefore, a unique solution of system (4.2.44) has the following form: N ."/ j ¥0 ./

."/ D ŒI  j PN ."/ ./1 pN 0 ./:

(4.2.45)

Taking into account formulas (4.2.32) and (4.2.45) we get the following formula for .0/ every j 2 X1 ,   0, and "  "0 defined in the proof of Lemma 4.2.1: N ."/ ./ j i0

D

N X

."/ Ei ıNj."/ ./pNk0 ./; k

i ¤ 0:

(4.2.46)

kD1 .0/

Relation (4.2.41), condition T001 , and formula (4.2.46) imply that, for every j 2 X1 and   0, ."/ 0 N ij ./

.0/ ! 0 Nij ./ as " ! 0;

i ¤ 0:

(4.2.47)

This relation is equivalent to the weak convergence relation .ˇ/ given in Lemma 4.2.3. The weak convergence relation ./ in Lemma 4.2.3 follows from relation (4.2.4) and the weak convergence relations .˛/ and .ˇ/ in this lemma. Remark 4.2.1. It follows from the proof of Lemma 4.2.3 that the assumption of the ."/ .0/ ./ ) Fi0 ./ as " ! 0, i ¤ 0, is not required in claim .˛/ of weak convergence Fi0 this lemma. Lemmas 4.2.1–4.2.3 allow us to also weaken condition I3 and replace it with the following conditions: .0/

.0/

I4 : Fj .0/ < 1 for some state j 2 X1 ; and I5 : F0.0/ .0/ < 1. ."/

.0/

Condition T1 implies that Fi ./ ) Fi ./ as " ! 0, i 2 X, and thus there exists ."/ ."/ "1 > 0 such that Fi .0/ < 1 and F0 .0/ < 1 for "  "1 . As was pointed out in Lemma 4.2.2, conditions A2 , T1 , E2 imply that any state .0/ j 2 X1 , for "  "0 , is a recurrent-without-absorption state for the semi-Markov processes ."/ .t /, t  0. Therefore, conditions A2 , T1 , E2 , and I4 , I5 imply that condition I2 holds for the semi-Markov processes ."/ .t /, t  0, for "  min."0 ; "1 /. Note also that condition I5 is not restrictive at all. It automatically holds for the modified semi-Markov process obtained by removing the absorption state, as described in Subsection 4.2.3. Thus, the use of this condition can be omitted in the corresponding asymptotic relations for non-absorption events.

186

4 Perturbed semi-Markov processes

4.2.5 Convergence of expectations for the first hitting times We denote Z ."/ mij D

1

0

."/ sFij .ds/;

."/ pij Œ1

Z D

1

."/

."/ ."/

sQij .ds/ D mij pij ;

0

i ¤ 0;

j 2 X:

Asymptotic results that describe transition phenomena in perturbed semi-Markov processes also involve the following moment perturbation condition: ."/

.0/

M11 : pij Œ1 ! pij Œ1 < 1 as " ! 0; i ¤ 0; j 2 X. Condition T1 (a) and the following condition obviously imply condition M11 to hold: ."/

.0/

M011 : mij ! mij < 1 as " ! 0, i ¤ 0, j 2 X. It is useful to note that the asymptotic relations in conditions M11 and M011 taken for fixed i ¤ 0, j 2 X are equivalent if the the asymptotic relation in condition T1 (a) .0/ .0/ holds for the same i and j and pij > 0. However, if pij D 0 then the asymptotic relation in condition M11 may hold while the corresponding relation in condition M011 does not. We introduce the following notations: Z

."/

0 Mij D ."/ jMi0

."/

."/

."/

."/

."/

."/

s 0 Gij .ds/ D Ei j .j

0

Z D

1

1

."/

i; j ¤ 0;

."/

i; j ¤ 0;

< 0 /;

sj Gi0 .ds/ D Ei j .0 < j /;

0

and ."/ Mij

Z D

0

1

."/

."/

."/

."/

."/

sGij .ds/ D 0 Mij C jMi0 D Ei .j ^ 0 /;

i; j ¤ 0: ."/

.0/

Lemma 4.2.4. Let conditions A2 , T1 , M11 , and E2 hold. Then .˛/ 0 Mij ! 0 Mij < .0/

."/

.0/

.0/

1 as " ! 0, i ¤ 0, j 2 X1 ; .ˇ/ jMi0 ! jMi0 < 1 as " ! 0; i ¤ 0; j 2 X1 ; ."/ .0/ .0/ ./ Mij ! Mij < 1 as " ! 0, i ¤ 0, j 2 X1 . Proof. Let us first note that condition M11 implies that there exists "2 > 0 such that sup max

""3 i¤0

X j 2X

."/ pij Œ1  M < 1:

(4.2.48)

187

4.2 Perturbed semi-Markov processes .0/

Using this estimate we get for every i ¤ 0, j 2 X1 , and "  "2 that ."/

Mij."/

."/

j ^0

X

D Ei

n."/

D Ei

nD1

D

1 X

X

1 X

n."/ .j."/ ^ 0."/ > n  1/

nD1

Ek 1."/ Ei .j."/ ^ 0."/ > n  1; ."/ n1 D k/

nD1 k¤j;0

M

1 X

."/

."/

Pi fj ^ 0 > n  1g  M

nD1

1 X

LŒn=m D

nD0

Mm < 1: (4.2.49) 1L

."/

The expectations 0 Mij , i ¤ 0, satisfy the following system of linear equations analogous to (4.2.8) for every j ¤ 0: ."/ 0 Mij

."/

D Lij C

X

."/

."/

pik 0 Mkj ;

i ¤ 0;

(4.2.50)

k¤j;0

where

."/ L."/ ij D pij Œ1 C

X

."/ ."/ pik Œ1 0 fkj ;

i; j ¤ 0:

k¤j;0 ."/ appear in the expression for the Note that there are the hitting probabilities 0 fkj

."/ ."/ ."/  free terms L."/ ij , because we operate with the expectations 0 Mij D Ei j .j

0."/ / of the first hitting times j."/ multiplied by the indicators of the hitting events

.j."/  0."/ /. For every j ¤ 0, system (4.2.50) can be rewritten in the following matrix form: ."/ 0 Mj

where

2 ."/ 0 Mj

6 D6 4

."/

."/

D Lj

C j P."/ 0 Mj ;

3

2

."/ 0 M1j

7 :: 7; : 5 ."/ 0 MNj

."/

Lj

(4.2.51)

3 ."/ L1j 6 : 7 : 7 D6 4 : 5: L."/ Nj

This system has the same coefficient matrix I  j P."/ for every j 2 X1.0/ as system (4.2.9). As was shown in the proof of Lemma 4.2.10, this matrix has an inverse for "  "0 , defined in formula (4.2.14). Define "3 D min."0 ; "2 /:

(4.2.52)

188

4 Perturbed semi-Markov processes .0/

A unique solution of system (4.2.51) has the following form for j 2 X1 "  "3 : ."/ 0 Mj

."/

D ŒI  j P."/ 1 Lj :

and

(4.2.53)

Taking into account (4.2.21) and (4.2.53) we get the following formula for every j 2 X1.0/ and "  "3 : ."/

0 Mij D

N X

."/

."/

Ei ıj k Lkj ;

i ¤ 0:

(4.2.54)

k; j ¤ 0:

(4.2.55)

kD1

By Lemma 4.2.1 and relation (4.2.25), ."/

.0/

Lkj ! Lkj as " ! 0;

Relation (4.2.55), conditions T1 (a), M11 , and formula (4.2.54) imply the convergence relation .˛/ in Lemma 4.2.4. ."/ The expectations jMi0 , i ¤ 0, for every j ¤ 0, satisfy the following system of linear equations that are similar to (4.2.50): ."/ jMi0

."/

D Lj 0 C

X

pik jMk0 ;

."/

."/

i ¤ 0;

."/

."/

i; j ¤ 0:

(4.2.56)

k¤j;0

where

."/

X

."/

Li0 D pi0 Œ1 C

pik Œ1 jfk0 ;

k¤j;0

For every j ¤ 0, system (4.2.56) can be rewritten in a matrix form, ."/ j M0

where

2 ."/ j M0

6 D6 4

."/

."/

D L0 C j P."/ j M0 ;

."/ jM10

:: :

."/ jMNj 0

3 7 7; 5

(4.2.57)

."/ 3 L10 7 6 D 4 ::: 5 :

2

."/

L0

."/

LN 0

.0/

For every j 2 X1 , this system has the same coefficient matrix I  j P."/ as system (4.2.9). As was shown in the proof of Lemma 4.2.10, this matrix has an inverse for "  "0 . Therefore, a unique solution of system (4.2.51) has the following form for .0/ j 2 X1 and "  "3 that are defined in formula (4.2.52): ."/ j M0

D ŒI  j P."/ 1 L."/ 0 :

(4.2.58)

189

4.2 Perturbed semi-Markov processes .0/

Taking into account formulas (4.2.21) and (4.2.58) we get for every j 2 X1 "  "3 that ."/

jMi0 D

N X

."/

."/

and

i ¤ 0:

(4.2.59)

k ¤ 0:

(4.2.60)

Ei ıj k Lk0 ;

kD1

By Lemma 4.2.1 and relation (4.2.25), we have ."/

.0/

Lk0 ! Lk0 as " ! 0;

Relation (4.2.60), conditions T1 (a), M11 , and formula (4.2.59) imply the convergence relation .ˇ/ in Lemma 4.2.4. Relation ./ in Lemma 4.2.4 is a direct corollary of relations .˛/ and .ˇ/ given in this lemma. Remark 4.2.2. It follows from the proof of Lemma 4.2.4 that the assumption that ."/ .0/ mi0 ! mi0 as " ! 0, i ¤ 0, is not required in claim .˛/ of this lemma.

4.2.6 Convergence of generalised hitting and absorption probabilities In what follows, we will also need to consider more general hitting and absorption probabilities. Let us set consider the set Z f1; : : : ; N g, and introduce the first hitting time in ."/ ."/ ."/ the set Z for the imbedded Markov chain n by Z D min.n  1 W n 2 Z/, and then define the hitting probabilities ."/ 0 fiZr

."/

."/

."/

D Pi fZ < 0 ;  ."/ .Z/ D j g;

i ¤ 0;

r 2 Z;

and the absorption probabilities ."/ Zfi0

."/ D Pi f0."/ < Z g; i ¤ 0: .0/

Lemma 4.2.5. Let conditions A2 , T1 (a), E2 hold, and Z \ X1 ¤ ¿. Then .˛/ ."/ .0/ ."/ .0/ 0 fiZr ! 0 fiZr as " ! 0, i ¤ 0, r 2 Z; .ˇ/ Zfi0 ! Zfi0 as " ! 0, i ¤ 0. Proof. Without loss of generality, we can assume that Z D f1; : : : ; mg. ."/ ."/ The case m D N is trivial. Indeed, by the definition, Z ^0 D 1 with probability ."/ ."/ ."/ ."/ 1 in this case and, therefore, 0 fiZr D pir , Zfi0 D pi0 , i ¤ 0, r 2 Z. ."/ Thus, we assume that m < N . The probabilities 0 fiZr , i ¤ 0, satisfy the following system of linear equations for Z such that Z \ X1 ¤ ¿ and every r 2 Z: ."/

."/

0 fiZr D pir C

N X kDmC1

."/

."/

pik 0 fkZr ;

i ¤ 0:

(4.2.61)

190

4 Perturbed semi-Markov processes

It is important that the first m equations in this system are just formulas that express ."/ ."/ , i D 1; : : : ; m, as linear combinations of the quantities 0 fiZj , the quantities 0 fiZj i D m C 1; : : : ; N . The last N  m equations make a determined system of linear equations for the latter quantities. Thus, N m is the real dimension of system (4.2.61). Systems (4.2.61) can be rewritten in the following matrix form: ."/ 0 fZr

where

2 ."/ 0 fZr

6 D4

."/ ."/ D p."/ r C Z P 0 fZr ;

."/ 0 f1Zr

:: :

3 7 5;

."/ 3 p1r 7 6 D 4 ::: 5 ;

2

p."/ r

."/ 0 fN Zr

and

(4.2.62)

."/

pN r

2

ZP

."/

3 ."/ ."/ 0 : : : 0 p1.mC1/ : : : p1N 6 : :: :: :: 7 7 : D6 : : : 5: 4 : ."/ ."/ 0 : : : 0 pN.mC1/ : : : pN N

Let us show that the coefficient matrix of this system, I  Z P."/ , has an inverse for every r 2 Z; Z \ X1.0/ ¤ ¿, and all " small enough, and, hence, a unique solution of systems (4.2.62) takes the following form: ."/ 0 fZr

D ŒI  Z P."/ 1 p."/ r :

(4.2.63) .0/

Indeed, take some recurrent-without-absorption state j 2 Z \ X1 . By the definition, the elements of the matrices Z P."/ and j P."/ satisfy the inequalities ."/

."/

0  .k  m C 1/pik  .k ¤ j /pik ;

i; k ¤ 0;

(4.2.64)

and, thus, the corresponding norms satisfy the following relation for every j 2 Z \ X1.0/ and "  "0 that are defined in formula (4.2.14): j .Z P."/ /n j  j .j P."/ /n j  LŒn=m ! 0 as n ! 0:

(4.2.65)

.0/

Relation (4.2.65) implies that for every r 2 Z, Z \ X1 ¤ ¿, there exists the inverse ŒI  Z P."/ 1 given by the following matrix series that converges in the norm: ŒI  Z P."/ 1 D I C Z P."/ C .Z P."/ /2 C    :

(4.2.66)

191

4.2 Perturbed semi-Markov processes

Let us introduce the random variables ."/

."/ ıZ k

."/

Z ^0

X

D

.."/ n1 D k/;

k ¤ 0:

nD1

By the definition, ."/ Ei ıZ k

D

1 X

."/

."/

."/

Ei .n1 D k; Z ^ 0 > n  1/:

(4.2.67)

nD1 ."/

Finally, comparing formulas (4.2.66) and (4.2.67) we get that Ei ıZ k < 1, i; k ¤ 0, Z \ X1 ¤ ¿, for every "  "0 , and that ."/

ŒI  Z P."/ 1 D k Ei ıZ k k:

(4.2.68)

It is useful to note that, by the definition, ."/

Ei ıZ k D .i D k/;

i; k 2 Z:

(4.2.69)

Condition T1 (a) implies in an obvious way that ZP

."/

! Z P.0/ as " ! 0:

(4.2.70)

As was mentioned above, the inverse matrix ŒI  Z P."/ 1 exists for all " small enough. Moreover, the elements of the inverse matrix ŒI  Z P."/ 1 are continuous rational functions of elements of the matrix I  Z P."/ . These rational functions have the non-zero denominator det.I  Z P."/ /. This implies that ŒI  Z P."/ 1 ! ŒI  Z P.0/ 1 as " ! 0:

(4.2.71)

Formulas (4.2.63) and (4.2.68) imply that, for every r 2 Z; Z \ X1.0/ ¤ ¿, and "  "0 , ."/ 0 fiZr

D

N X

."/

."/

Ei ıZ k pkr ;

i ¤ 0:

(4.2.72)

kD1 .0/

Using relations (4.2.68) and (4.2.71) we see that, if Z \ X1 ."/ .0/ ! Ei ıZ as " ! 0; Ei ıZ k k

i; k ¤ 0:

¤ ¿, then (4.2.73)

Relation (4.2.73), condition T1 (a), and formula (4.2.72) imply the convergence relation .˛/ given in Lemma 4.2.5.

192

4 Perturbed semi-Markov processes .0/

."/

For every Z such that Z \ X1 ¤ ¿, the probabilities Zfi0 , i ¤ 0, satisfy the following system of linear equations that are similar to (4.2.61): ."/

N X

."/

Zfi0 D pi0 C

."/

pik

."/ Zfk0 ;

i ¤ 0:

(4.2.74)

kDmC1

System (4.2.74) can be rewritten in the following matrix form: ."/ Z f0

where

2 ."/ Z f0

6 D4

."/ ."/ D p."/ 0 C Z P Z f0 ;

."/ Zf10

3

(4.2.75)

."/ 3 p10 7 6 D 4 ::: 5 :

2

7 :: 5; : ."/ ZfN 0

p."/ 0

."/

pN 0

This system has the same coefficient matrix I  Z P."/ as system (4.2.62). As was shown above, this matrix has an inverse for every Z such that Z \ X1.0/ ¤ ¿ and "  "0 . Therefore, a unique solution of system (4.2.75) has the following form: ."/ Z f0

."/

D ŒI  Z P."/ 1 p0 :

(4.2.76)

Taking into account formulas (4.2.68) and (4.2.76) we get the following formula for .0/ every Z such that Z \ X1 ¤ ¿ and "  "0 defined in (4.2.14): ."/

Zfi0 D

N X

."/

."/

Ei ıZ k pk0 ;

i ¤ 0:

(4.2.77)

kD1

Relation (4.2.73), condition T1 (a), and formula (4.2.77) imply the convergence relation .ˇ/ in Lemma 4.2.5.

4.3 Asymptotics of cyclic absorption probabilities In this section we give a description of asymptotics for cyclic absorption probabilities for perturbed semi-Markov processes.

4.3.1 Stationary distribution for the imbedded Markov chain Let us assume that the following two conditions hold for the limit imbedded Markov .0/ chain n : P .0/ A3 : .0/ p i0 D 0. i2X 1

193

4.3 Asymptotics of cyclic absorption probabilities

Conditions A3 and E02 are equivalent to the following condition: .0/

E000 2 : X1 D f1; : : : ; N g is a closed communicative class of states for the limit imbed.0/ ded Markov chain n , n D 0; 1; : : : . .0/

Let introduce a random variable &i .n/ which is the number of visits of the Markov .0/ chain n in the state i during the first n transitions, &j.0/ .n/ D

n X

..0/ r 1 D j /;

n D 0; 1; : : : ;

i 2 X1.0/ :

r D1

The following formula follows directly from the definition of the random variables .0/ &i .n/: .0/

Ei &i .n/ .0/ D pNij .n  1/ n Pn .0/ r D1 pij .r  1/ D ; n where

.0/ pij .n/ D Pi f.0/ n D j g;

n D 0; 1; : : : ;

.0/

i; j 2 X1 ;

(4.3.1)

i; j 2 X1.0/ :

n D 0; 1; : : : ;

The statement in the lemmas given below can serve to define a stationary distribu.0/ .0/ tion %.0/ i , i 2 X1 , for the closed communicative class of states X1 . .0/

.0/

Lemma 4.3.1. Let condition E000 2 hold. Then .˛/ Pi f&j .n/=n ! %j g D 1, i; j 2 P .0/ .0/ .0/ .0/ .0/ .0/ X1 , where (a) %j > 0, i 2 X1 , (b) j 2X .0/ %j D 1; .ˇ/ pNij .n1//=n ! %j 1

as n ! 1, i; j 2 X1.0/ . .0/

.0/

The stationary distribution %i , i 2 X1 can also be uniquely characterised as a solution of the following system of linear equations: X .0/ .0/ X .0/ .0/ %i D %k pki ; i ¤ 0; %k D 1: (4.3.2) .0/

k2X1

.0/

k2X1

It should be noted that the first N equations of system (4.3.2) are linearly dependent, since the sum of the quantities in the left and the right hand sides equals 1. Thus, in fact, there are only N linearly independent equations in this system. .0/

Lemma 4.3.2. Let condition E000 2 hold. Then .˛/ the stationary distribution %i , i 2 .0/ X1 , is a unique solution of the system of linear equations (4.3.2).

194

4 Perturbed semi-Markov processes

The following lemma supplies two useful formulas for the stationary distribution i 2 X1.0/ , that will be used in the sequel.

%.0/ i ,

.0/

Lemma 4.3.3. Let condition E000 2 hold. Then .˛/ %i .0/ .0/ .0/ .0/ Ei ıik D %k =%i , i; k 2 X1 .

.0/

.0/

D .Ei i /1 , i 2 X1 ; .ˇ/

.0/

Proof of Lemmas 4.3.1– 4.3.3. If one chooses the distribution functions Fij .t / D .0/

Œ1;1/ .t /, then the semi-Markov process .0/ .t / coincides with the Markov chain n .0/ in the sense that .0/ .t / D Œt , t  0. .0/

By applying Lemma 4.2.4 in this case, we get that (c) Ei j .0/

.0/

.0/

< 1, i; j 2 X1 .

Let us introduce the random variable j .n/ D min.k W &i .k/ D n/ that is the .0/

moment of the n-th hitting of the Markov chain n into the state j . As is known, .0/ .0/ the moments j .n/, n D 0; 1; : : :, are regeneration times for the Markov chain n , .0/

.0/

and, thus, the random variables j .n/  j .n  1/, n D 1; 2; : : :, are independent, .0/

moreover, these random variables have the same distribution Pj fj

 t g for n  2. .0/

Thus, by the standard strong law of large numbers, we have (d) Pi fj .n/=n ! .0/

.0/

.0/

.0/

Ej j g D 1, i; j 2 X1 . By the definition, &i .n/ D max.k W j .k/  n/, and, .0/

.0/

thus, relation (c) implies, in an obvious way, that (e) Pi f&j .n/=n ! .Ej j /1 g D .0/

.0/

.0/

1, i; j 2 X1 . It follows from (c) that .Ej j /1 > 0, j 2 X1 . Since, by P .0/ the definition, j 2X .0/ &j .n/=n D 1, n D 0; 1; : : :, relation (e) also implies that 1 P .0/ 1 D 1. Thus, relation .˛/ given in Lemma 4.3.1 is proved. Also .0/ .Ej  j / j 2X 1

.0/

.0/

.0/

we have proved that the stationary probabilities satisfy %j D .Ej j /1 , i 2 X1 , that is relation .ı/ in Lemma 4.3.3. .0/ Taking into account that, by the definition, the random variables j&j .n/=nj  1 .0/

.0/

are bounded, we get .˛/, and by the Lebesgue theorem, (f) Ei &j .n/=n ! %j

as

.0/ X1 .

Relation (f) is equivalent, due to (4.3.1), to the relation .ˇ/ n ! 1, i; j 2 P .0/ in Lemma 4.3.1. Now, let us use the relation (g) nrD1 pij .r  1/ D ı.i; j / C Pn1 .0/ .0/ P .0/ .0/ . r D1 pik /pkj , n  1, i; j 2 X1 . Dividing the expressions in the left k2X1 and the right hand-side of (g) by n and then making n tend to 1 we get the identities P .0/ (h) %j.0/ D k2X .0/ %.0/ pkj , j 2 X1.0/ . Thus, the stationary distribution is a solution k 1

.0/

.0/

of system (4.3.2). Let us assume that %Qj , j 2 X1 , is another solution of this system. Then, by iterating the equations in this system we get, in an obvious way, P .0/ .0/ .0/ .0/ .0/ that %Qj , j 2 X1 , should also satisfy the relations (k) %Qj D k2X .0/ %Q k pkj .n/, .0/

1

.0/

j 2 X1 , n  1. By summing up these relations, we get also the relations (l) n%Qj

D

195

4.3 Asymptotics of cyclic absorption probabilities

.0/ Pn .0/ .0/ %Q k r D1 pkj .r/, j 2 X1 , n  1. Dividing the expressions in the left and the right hand sides of (l) by n and then approach n to 1 we get the identities P .0/ .0/ .0/ .0/ .0/ (m) %Qj D Q k %j D %j , j 2 X1 . Here we also used that, by the .0/ % k2X

P

.0/ k2X1

1

.0/

.0/

assumption made above, %Qj , j 2 X1 , is a solution of system (4.3.2), and, therefore, P .0/ Q k D 1. Identities (m) prove that any solution of system (4.3.2) coincides .0/ % k2X1 with the stationary distribution. Thus, this distribution is a unique solution of system (4.3.2). It remains to prove the formulas given in Lemma 4.3.3. Let us introduce the ranP dom variable ıj.0/ .n/ D ..0/ D k/, n  1, j; k 2 X1.0/ . r k  .0/ .n1/r 0 automatically for all " small enough, (b) if there exists a sequence 0 < "n ! 0 as n ! 1 such that f ."n / D 0, n  1, then  D 0. Lemma 4.3.4. Let conditions A2 , A3 , T1 (a), and E02 hold. Then, .˛/ jfj."/ 0 ! 0 as ."/ ! =%.0/ as " ! 0, j 2 X1.0/ . Let also condition B6 holds. Then, .ˇ/ jfj."/ 0 t j

" ! 0, j 2 X1.0/ .

Proof. Statement .˛/ follows directly from Lemma 4.2.1 and condition A3 , which imply that ."/ jfj 0

.0/

! jfj 0 D 0 as " ! 0:

(4.3.7)

Let us prove statement .ˇ/. Obviously, condition A3 holds for the first hitting time 0.0/ D 1 a.s., and, thus, the following identity holds with probability 1: .0/

ıj.0/ k

.0/

j ^0

D

X

nD1



..0/ n1

D k/ D

.0/

j X

nD1

..0/ n1 D k/:

(4.3.8)

4.3 Asymptotics of cyclic absorption probabilities

197

Let us now use the well-known formula that connects the expected value of the number of visits of the imbedded Markov chain .0/ n into the state k, up to the first hitting into the state j , with the stationary probabilities .0/

.0/

.0/

.0/

Ej ıj k D %k =%j ;

j; k 2 X1 :

(4.3.9)

Relations (4.2.21) and (4.2.22) imply that ! Ej ıj.0/ as " ! 0; Ej ıj."/ k k

j; k 2 X1.0/ :

(4.3.10)

Let us first consider the case where  D 0 in condition B6 . ."/ .0/ .0/ Obviously, pk0  f ."/ =%k , k 2 X1 . Using these inequalities, relations (4.2.21), (4.2.22), and formula (4.3.3) we have X ."/ ."/ ."/ lim jfj 0  t ."/  lim Ej ıj k pk0 t ."/ "!0

"!0

.0/

k2X1

 lim .

X

"!0

."/

.0/

Ej ıj k =%k /  lim f ."/ t ."/ "!0

.0/

k2X1 .0/

 .N=%j /  lim f ."/ t ."/ D 0:

(4.3.11)

"!0

Let us now consider the case where  > 0 in condition B6 . As was mentioned above, f ."/ > 0 in this case for all " small enough, say 0 < "  "4 . Using relations (4.2.21), (4.2.25), and formulas (4.3.3), (4.3.9) we get for 0 < "  "4 that ."/

.0/

j jfj 0  f ."/ =%j .0/

f ."/ =%j

j





.0/ ."/

N X

%j pk0 j Ej ıj."/  %.0/ =%j.0/ j  P k k .0/ ."/ N iD1 %i pi0 kD1 N X

j

."/ Ej ıj k



.0/ .0/ %k =%j

kD1

j

%j.0/ %.0/ k

! 0 as " ! 0:

(4.3.12)

Relation (4.3.12) implies that ."/ .0/ jfj 0 %j

f ."/

! 1 as " ! 0;

(4.3.13)

and, thus, ."/ jfj 0

t

."/

D

."/ .0/ jfj 0 %j

The proof is complete.

f ."/



f ."/ t ."/ .0/ %j

.0/

! =%j

as " ! 0:

(4.3.14)

198

4 Perturbed semi-Markov processes

4.3.3 Asymptotics of cyclic absorption probabilities for semi-Markov processes with non-recurrent-without-absorption states An asymptotic analysis of cyclic absorption probabilities becomes more complicated in the case where condition E002 holds. ."/

.0/

Let conditions A2 , A3 , T1 (a), and E002 hold, and 0 D i 2 X1 . We de."/ fine subsequent return times by O ."/ .n/ D min.k >  ."/ .n  1/ W k 2 Y /, .0/

n D 1; 2; : : : ; O ."/ .0/ D 0, where Y D X1 [ f0g. In this case, as follows from Lemma 4.2.2, the random variables O ."/ .n/, n D 0; 1; : : : are finite with probability 1 for "  "0 . Thus, for "  "0 we can define the random sequence O ."/ .n/ D ."/ , O ."/ .n/ n D 0; 1; : : : . This sequence is a homogeneous Markov chain with the phase space Y . Obviously, .0/ 0 is an absorption state while X1 is one class of communicative states for this Markov chain. The transition probabilities of this Markov chain for non-absorbing states are given by the following formulas: ."/

X

."/

pOij D pij C

."/

pik 0 f

."/ .0/

kX1 j

.0/ k2X2

.0/

i; j 2 X1 ;

;

(4.3.15)

and ."/

."/

pOi0 D pi0 C

X .0/ k2X2

."/

."/

.0/

i 2 X1 :

pik X .0/ fk0 ; 1

(4.3.16)

."/

It follows from conditions A3 and T1 (a) that pik ! 0 as " ! 0 for the states

i 2 X1.0/ ; k 2 X2.0/ . Thus, ."/

.0/

.0/

pOij ! pOij D pij

as " ! 0;

.0/

i; j 2 X1 ;

(4.3.17)

i 2 X1.0/ :

(4.3.18)

and ."/ .0/ pOi0 ! pOi0 D 0 as " ! 0;

It is obvious that the following identity holds: O."/ jfj 0

."/

."/

D Pj fO 1 D 0g D jfj 0 ;

.0/

j 2 X1 :

(4.3.19) ."/

Thus the cyclic absorption probabilities coincide for the Markov chains n and ."/ O n . Also, relations (4.3.17) and (4.3.18) imply that conditions A2 , T1 (a), and E02 hold ."/ for the Markov chain O n .

199

4.3 Asymptotics of cyclic absorption probabilities

Denote

X

fO."/ D

.0/ ."/

%i pOi0 :

i2X1.0/

Conditions A3 , T1 (a), and E002 evidently imply that fO."/ ! fO.0/ D 0 as " ! 0:

(4.3.20)

Let us also assume the following condition that relates the rate of growth of time, ! 1, and the speed of the perturbation determined, in this case, by the rate with which the averaged absorption probability vanishes, fO."/ ! 0:

t ."/

B7 : 0  t ."/ ! 1 and fO."/ ! 0 as " ! 0 in such a way that fO."/ t ."/ ! , where 0    1. Let us remark that condition B7 implies the following: (c) if  > 0 then automatically fO."/ > 0 for all " small enough, (d) if there exists a sequence 0 < "n ! 0 as n ! 1 such that fO."n / D 0; n  1, then  D 0. Finally, we can get an asymptotics for the cyclic absorption probabilities jfj."/ 0 ,

j 2 X1.0/ by applying Lemma 4.3.4 to the Markov chains O ."/ n .

."/

Lemma 4.3.5. Let conditions A2 , A3 , T1 (a), and E002 hold. Then, .˛/ jfj 0 ! 0 as

."/ ! =%.0/ as " ! 0, " ! 0, j 2 X1.0/ . Let also condition B7 hold. Then .ˇ/ jfj."/ 0 t j

j 2 X1.0/ .

Lemmas 4.3.4 and 4.3.5 play an important role in what follows. They make it possible to formulate the corresponding balancing conditions between the time and the absorption probabilities in a form invariant with respect to the choice of the state used to build the regeneration cycles.

4.3.4 Balancing conditions for cyclic absorption probabilities In conclusion, let us formulate a condition under which B6 implies that B7 holds with the same function t ."/ and the limit . By using formulas (4.2.70), (4.2.77) in the case where Z D X1.0/ and then formula (4.3.16), we can represent the quantity fO."/ in the following form: X .0/ ."/ X ."/ ."/ ."/ fO."/ D %i .pi0 C pik X .0/ fk0 / D f ."/ C f1 ; (4.3.21) .0/

where

."/

f1

1

.0/

i2X1

k2X2

D

X .0/

i2X1

.0/

%i

X .0/

k2X2

."/

pik

X .0/

r2X2

Ek ı

."/ .0/

X1

."/

r

pr 0 :

200

4 Perturbed semi-Markov processes ."/

.0/

Conditions T1 (a) and E002 imply that (e) pik ! 0 as " ! 0, i 2 X1 , k 2 X2 , and (f) Ek ı

."/

.0/

X1

r

! Ek ı

."/

.0/

X1

.0/

< 1 as " ! 0, k; r 2 X2 .

r

It is useful to note that relations (e) and (f) imply that (g) f1."/ ! 0 as " ! 0, and, ."/ .0/ therefore, (h) fO."/ ! 0 as " ! 0 even if (i) pk0 6! 0 as " ! 0 for k 2 X2 . Let us assume the following condition: ."/ ."/ t < 1, k 2 X2.0/ . B8 : lim"!0 pk0

If conditions B6 and B8 hold, then formula (4.3.21) and relations (e) and (f) above ."/ easily give that (h) f1 t ."/ ! 0 as " ! 0. Therefore, condition B7 holds with the same function t ."/ and the limit  as in condition B6 , i.e., lim fO."/ t ."/ D lim f ."/ t ."/ D :

"!0

"!0

(4.3.22)

If condition B8 does not hold, in particular, if f ."/ D 0 for " > 0, then the asymptotics for the quantities fO."/ and, as a consequence, the cyclic absorption probabilities ."/ jfj 0 can be more complicated and require an additional careful analysis. We will consider examples of such models in Chapter 5.

4.4 Mixed ergodic and limit theorems for perturbed semi-Markov processes In this section we present mixed ergodic and limit theorems for perturbed regenerative processes. These theorems describe pseudo-stationary phenomena for nonlinearly perturbed semi-Markov processes.

4.4.1 Non-arithmetic properties of distributions of return times Let us formulate now a condition that would guarantee that the limit distribution functions of the return times Gi.0/ i .t / are arithmetic. A distribution function F .x/ is called generalised arithmetic if there exists a positive h and a real constant d such that the following relation holds: 1 X

.F .nh  d /  F .nh  d  0// D 1:

(4.4.1)

nD1

Denote by Uh D f0; ˙h; ˙2h; : : :g the lattice of points with step h > 0. Let also Ii.m/ be, for i 2 X1.0/ , m  1, the set of all cyclic chains of states .i0 ; : : : ; im / such that (a) i0 D i , i1 ; : : : ; im1 ¤ i; 0 and i1 ; : : : ; im1 2 X1.0/ , im D i ; Q .0/ (b) m nD1 pin1 in > 0.

4.4 Mixed ergodic and limit theorems for perturbed semi-Markov processes

201

Let us introduce the following condition: N 1 : There exist positive h and real constants dr k , r; k 2 X .0/ , satisfying the followN 1 ing: P1 .0/ .0/ .0/ (a) nD1 .Fr k .nh  dr k /  Fr k .nh  dr k  0// D 1 for any r; k 2 X1 , such that pr.0/ > 0; k Pm .m/ for (b) nD1 din1 in 2 Uh for any cyclic chain of states .i0 ; : : : ; im / 2 Ii .0/ every i 2 X1 and m  1.

Denote

N .0/ 0 Gi i .t /

.0/

.0/

D 0 Gi i .t /=0 Gi i .1/;

t  0;

.0/

i 2 X1 :

.0/ .0/ Note that condition E2 implies that 0 Gi.0/ i .1/ D 0 fi i > 0 for i 2 X1 . .0/ Lemma 4.4.1. Let conditions A2 and E2 hold. Then .˛/ if 0 GN i i .t / is an arithmetic .0/ distribution function for some i 2 X1 , then it is an arithmetic distribution function .0/ N 1 is necessary and sufficient for the distribution for every i 2 X1 ; .ˇ/ condition N .0/ .0/ functions 0 GN i i .t /, i 2 X1 , to be arithmetic.

Proof. Denote Z 1 Z .0/ .0/ is .0/ e Fij .ds/; 0 'ij ./ D ij ./ D 0

1 0

.0/ e is 0 GN ij .ds/;  2 R1 ; i; j 2 X:

.0/ Let us assume that the distribution function 0 GN i i .t / is arithmetic. As is known .0/ (see, for example, Feller (1971)) this is so if and only if (a) 0 'i i .h / D 1 for some P1 .0/ .0/ h D 2= h > 0 or, equivalently, (b) nD1 .0 GN i i .nh/  0 GN i i .nh  0// D 1. Obviously, .0/ .0/ 1 0 'i i ./ D .0 fi i /

1 X

m Y

X

mD1 .i ;:::;i /2I .m/ 0 m i

nD1

1 X

m Y

.0/ .0/ in1 in ./pin1 in ;

 2 R1 ; (4.4.2)

and, thus, .0/

.0/

j0 'i i ./j  .0 fi i /1

X

mD1 .i ;:::;i /2I .m/ nD1 0 m i

j

.0/ .0/ in1 in ./jpin1 in ;

 2 R1 : (4.4.3)

Note that, by the definition, .0/ 0 fi i D

1 X

X

m Y

mD1 .i ;:::;i /2I .m/ nD1 0 m i

pi.0/ : n1 in

(4.4.4)

202

4 Perturbed semi-Markov processes

It follows from relations (4.4.3) and (4.4.4) that (c) j

.0/ .h /j rk

.0/

D 1 if pr k > 0.

Indeed, since X1.0/ is a communicative class of states, for r; k 2 X1.0/ such that P pr.0/ > 0 there exists a chain of states .i0 ; : : : ; im / 2 Ii.m/ such that m nD1 ..in1 ; k in / D .r; k// > 0. Thus, if (c) would not hold, i.e., j r; k 2

X1.0/

j 0 'i.0/ i .h /j

such that

< 1 for some

> 0, then relations (4.4.3) and (4.4.4) would imply that

< 1.

Relation (d) j ih drk

pr.0/ k

.0/ .h /j rk

.0/ .h /j rk

D 1 means that there exists real dr k such that .0/ ./e idrk rk

.0/ .h / rk

D

e . In this case, the characteristic function takes value 1 in point h . Thus, by the same result as mentioned above, relation (d) is equivalent to the P .0/ .0/ following relation: (e) 1 nD1 .Fr k .nhdr k /Fr k .nhdr k 0// D 1. Therefore, N 1 (a) holds. condition N Relation (4.4.2), taken for  D h , can be re-written in the following form: 1 1 D .0 fi.0/ i /

1 X

X

mD1 .i ;:::;i /2I .m/ 0 m i

expfi h

m X

din1 in g

nD1

m Y nD1

pi.0/ : n1 in

(4.4.5)

P It follows from (4.4.4) that this equality holds if and only if h m nD1 din1 in 2 U2 Pm or, equivalently, (f) nD1 din1 in 2 Uh for any chain of states .i0 ; : : : ; im / 2 Ii.m/ for every m  1. P Let us now prove that (f) implies that (g) m nD1 din1 in 2 Uh for any chain of states .m/ .i0 ; : : : ; im / 2 Ij for every m  1 and any j ¤ i , j 2 X1.0/ . .m/

Take an arbitrary chain of states .i0 ; : : : ; im / 2 Ij . Two cases are possible for this chain: (h0 ) i1 ; : : : ; im1 ¤ i , and (h00 ) there exists a subsequence 0 < n1 <    < nk < m such that in1 ; : : : ; ink D i while in ¤ i , 0 < n < m, n ¤ n1 ; : : : ; nk .

Let us first consider the case (h0 ). Since X1.0/ is a class of communicative states, .0/ (i) there exist two sequences of states r0 D i , r1 ¤ i; j , r1 2 X1 ; : : : ; rq1 ¤ i , .0/ .0/ j; jq1 2 X1 , jq D j and j0 D j , j1 ¤ j; i , j1 2 X1 ; : : : ; jl1 ¤ j; i; jl1 2 Q Q .0/ q X1 ; jl D i such that nD1 prn1 rn > 0 and lnD1 pjn1 jn > 0. Then (h) and (i) imply that (j) two chains of states .r0 ; : : : ; rq1 ; i0 ; : : : ; im1 ; j0 ; : : : ; jl / and .r0 ; : : : ; .qCmCl/ rq1 ; i0 ; : : : ; im1 ; i0 ; : : : ; im1 , j0 ; : : : ; jl / belong, respectively, to the sets Ii P P P .qC2mCl/ q l and Ii . Thus, (k) nD1 drn1 rn C m nD1 din1 in C nD1 djn1 jn 2 Uh and Pm Pl Pq nD1 din1 in C nD1 drn1 rn C 2 P nD1 djn1 jn 2 Uh . Obviously, (k) implies that (l) the difference of these sums is m nD1 din1 in 2 Uh , i.e., (g) holds for the chain of states .i0 ; : : : ; im /. Let us now consider the case (h00 ). In this case, the chains of states satisfy .n n / .n n / .in1 ; : : : ; in2 / 2 Ii 2 1 ; : : : ; .ink1 ; : : : ; ink / 2 Ii k k1 and .ink ; : : : ; im1 ; P P k .mnk Cn1 / 2 i0 ; : : : ; in1 / 2 Ii . Thus, (m) nnDn din1 in C    C nnDn din1 in 1 C1 k1 C1

4.4 Mixed ergodic and limit theorems for perturbed semi-Markov processes

203

P Pm C m nDnk C1 din1 in D nD1 din1 in > 0, i.e., again (g) holds for the chain of states .i0 ; : : : ; im /. N 1 (b) also holds. Proposition (g) implies that condition N .0/ Thus, (n) the assumption that the distribution 0 GN i i .t / is arithmetic for some i 2 .0/ N 1 holds. X1 implies that condition N N 1 (a) implies that .0/ .h /e ih drk D 1 if On the other hand, (o) condition N rk .0/ N 1 (b) implies that relation (4.4.5) holds. Using (4.4.4), p > 0, while (p) condition N rk

.0/

(o) and (p) we get for any i 2 X1

that

.0/ 0 'i i .h / .0/

D .0 fi i /1

1 X

X

m Y

mD1 .i ;:::;i /2I .m/ nD1 0 m i .0/

D .0 fi i /1

1 X

X

expfih

mD1 .i ;:::;i /2I .m/ 0 m i 1 D .0 fi.0/ i /

1 X

X

.0/ ih din1 in ih din1 in .0/ e pin1 in in1 in .h /e m X

din1 in g

nD1 m Y

mD1 .i ;:::;i /2I .m/ nD1 0 m i

m Y nD1

.0/

pin1 in

pi.0/ D 1: n1 in

(4.4.6)

As was mentioned above, the equality 0 'i.0/ i .h / D 1 for some h > 0 implies that .0/ N the distribution function Gi i .t / is arithmetic. N 1 implies that distribution 0 GN .0/ .t / is Thus we have proved that (q) condition N ii .0/ arithmetic for any i 2 X1 . Statements (n) and (q) prove both propositions of Lemma 4.4.1. N 1: In fact, we will use the following condition that is opposite to condition N .0/

N1 : There does not exist positive h and real constants dr k , r; k 2 X1 such that both N 1 (a) and N N 1 (b) hold. conditions N According to Lemma 4.4.3, condition N1 is necessary and sufficient for the distri.0/ .0/ bution functions of return times, 0 Gi i .t /, i 2 X1 , to be non-arithmetic. Condition N1 is not restrictive. It holds, for example, if the distribution function .0/ .0/ Fr k .t / has an absolutely continuous component for some r; k 2 X1 such that

pr.0/ > 0. k It is also useful to mention that condition N1 automatically implies condition I4 (a). .0/ .0/ Indeed, if this condition would not hold, i.e., Fi .0/ D 1, i 2 X1 , then it is obvious .0/ .0/ .0/ .0/ that 0 Gi i .0/ D 1, i 2 X1 , i.e., the distribution functions Gi i .t /, i 2 X1 , would be concentrated in zero. This would contradict to the non-arithmetic property of these distribution functions.

204

4 Perturbed semi-Markov processes

4.4.2 Mixed ergodic and limit theorems for perturbed semi-Markov processes without non-recurrent-without-absorption states Now we are in a position to formulate basic results concerning pseudo-stationary phenomena for perturbed semi-Markov processes. We assume that conditions A2 , T1 , M11 , and A3 hold. Let us first consider the case where condition E02 holds. Denote .0/

j.0/ D P where

.0/

mi

X

D

.0/

%j mj .0/ i2X1

.0/ .0/ %i mi

.0/

pik Œ1 D

k2X

X

;

j 2 X1.0/ ;

.0/ .0/

mik pik ;

(4.4.7)

i ¤ 0:

k2X .0/

.0/ .0/

Note that here and henceforth the product pik Œ1 D mik pik should be counted .0/

.0/

.0/

as 0 if mik D 1 but pik D 0. In particular, condition A3 implies that pi0 Œ1 D .0/ .0/

mi0 pi0 D 0. .0/ .0/ As is known, j , j 2 X1 , is a stationary distribution for the limit semi-Markov .0/

process .0/ .t /, t  0, for the class of states X1 . Let us also denote X .0/ .0/ m.0/ D %i mi : i2X1.0/

Theorem 4.4.1. Let conditions A2 , T1 , E02 , M11 , A3 , N1 hold, and 0  t ."/ ! 1 as " ! 0 such that condition B6 holds. Then ."/ Pij."/ .t ."/ / D Pi f."/ .t ."/ / D j; ."/ g 0 >t .0/

! j e =m

.0/

as " ! 0;

i; j ¤ 0:

(4.4.8)

Proof. According to the remarks made in Subsection 4.2.1, the semi-Markov process ."/ .t /, more precisely, its modified version with the absorption state removed according to the procedure described in that subsection, can be considered as a perturbed regenerative process. The absorption time ."/ 0 is regarded as the regenerative stopping time. .0/ Let us chose some state j 2 X1 . Note that, according to condition E02 , we have .0/ .0/ X1 D X0 D f1; : : : ; N g. We can apply Theorem 3.2.1 to prove the asymptotic relation (4.4.8) in the case where i D j , since, in this case, ."/ .t / is a standard regen."/ erative process with regenerative times j .n/ that are subsequent times of return of the process ."/ .t / into the state j .

4.4 Mixed ergodic and limit theorems for perturbed semi-Markov processes

205

The renewal equation (3.2.1) now takes the form of renewal equation (4.2.1). In this case, A D fj g and the probability Pjj."/ .t / plays the role of the probability P ."/ .t; A/. ."/

The distribution function 0 Gjj .t / is regarded as the distribution function F ."/ .t / that generates this renewal equation. Conditions A2 , T1 , E02 , A3 , and N1 imply, by Lemmas 4.2.1, 4.2.3, 4.3.4, and 4.4.1, respectively, that (k) the distribution functions ."/ .0/ .0/ 0 Gjj .t / ) 0 Gjj .t / as " ! 0, (l) the limit distribution function 0 Gjj .t / is proper, and (m) it is also non-arithmetic. It follows from (k)–(m) that the condition of weak ."/ convergence D13 holds for the distribution functions 0 Gjj .t /. ."/

The distribution function Gjj .t / is regarded here as the distribution function

FO ."/ .t /. Conditions A2 , T1 , E02 , and M11 imply, by Lemmas 4.2.3 and 4.2.4, (n) ."/ .0/ the distribution functions 0 Gjj .t / ) 0 Gjj .t / as " ! 0, and (o) convergence of the ."/

.0/

first moments of these distribution functions, Mjj ! Mjj as " ! 0. The convergence relations (k) and (o) imply, by Remark 1.2.2, that (p) condition M8 holds for the ."/ distribution functions Gjj .t /. ."/

The function 1  Fj .t / is a forcing function for the renewal equation (4.2.1). ."/

The distribution functions Fj .t / are non-decreasing, and, by condition T1 , weakly ."/

converge. Thus, (q) the functions 1  Fj .t / converge locally uniformly in every .0/

continuity point of the limit function 1  Fj .t /. Indeed, denote by Cj the set of positive points of continuity of the distribution function Fj.0/ .s/. Take a point t 2 Cj and let sn , n D 0; 1; : : :, be a sequence of positive numbers such that (r) sn ! 0 as n ! 1 and (s) the points t ˙ sn are points .0/ of continuity of the distribution function Fj .s/ for all n D 0; 1; : : : . Using (r), (s), ."/

and weak convergence of Fj .t / we have that for any n D 0; 1; : : :, lim

lim

."/

.0/

sup j.1  Fj .t C v//  .1  Fj .t //j

u!0 0"!0 jvju

."/

.0/

D lim .j.1  Fj .t  sn //  .1  Fj .t C sn //j 0"!0

C j.1  Fj."/ .t C sn //  .1  Fj.0/ .t  sn //j/ .0/

.0/

 2j.1  Fj .t  sn //  .1  Fj .t C sn //j ! 0 as " ! 0:

(4.4.9)

It remains to note that the set C j is at most countable and, therefore, (t) C j has Lebesgue measure zero, m.C j / D 0. Thus condition F7 holds for the functions 1  ."/ Fj .t /. ."/

Finally, the probability 0 fjj can be regarded as the stopping probability f ."/ in one regeneration cycle. The balancing condition B6 implies, by Lemma 4.3.4, that

206 ."/ ."/ 0 fjj t

4 Perturbed semi-Markov processes .0/

! =%j

as " ! 0. Thus, the balancing condition B4 holds with the limit

.0/ =%j .

constant 0 D Thus, all conditions of Theorem 3.2.1 are satisfied. It remains to calculate the .0/ 0 parameters in the corresponding limit expression  .0/ .A/e  =m1 in the asymptotic relation (3.2.10) in this theorem. .0/ .0/ .0/ Due to conditions E02 and A3 , we have 0 fkj D 1, k; j 2 X1 and pi0 D 0, .0/ i 2 X1 . .0/ Therefore, the coefficients Lij take, in this case, the following form: .0/ .0/ L.0/ ij D mij pij C

m.0/ p .0/ ik ik

k¤j;0

X

D

X

.0/ .0/ mik pik

.0/

.0/

D mi ;

i; j 2 X1 :

(4.4.10)

.0/ k2X1

Thus, according to formulas (4.2.54), (4.3.9), and (4.4.10), X .0/ Ej ıj.0/ L.0/ 0 Mjj D k kj .0/

k2X1

D

X

.0/

.0/

.0/

.0/

.%k =%j /mk D m.0/ =%j ;

.0/

j 2 X1 ;

(4.4.11)

.0/ k2X1

and, therefore, .0/

=%j

.0/

0 =0 Mjj D

.0/ m.0/ =%j

D =m.0/ ;

.0/

j 2 X1 :

(4.4.12)

The corresponding limit stationary probability calculated with the use of formula (3.2.9) for the set A D fj g yields R1 .0/ 0 .1  Fj .s//ds .0/  .A/ D .0/ 0 Mjj D

mj.0/ m.0/ =%j.0/

D

%j.0/ mj.0/ m.0/

.0/

D j ;

.0/

j 2 X1 :

(4.4.13)

Now, by applying the asymptotic relation (3.2.10) given in Theorem 3.2.1, we get the asymptotic relation (4.4.8) for the case where i D j . We can apply Theorem 3.2.5 to prove the asymptotic relation (4.4.8) for i ¤ j . In ."/ this case, ."/ .t / is again a regenerative process with regeneration times j .n/ that are subsequent times of return of the process ."/ .t / into the state j . This process

4.4 Mixed ergodic and limit theorems for perturbed semi-Markov processes

207

has a transition period, because the initial state differs from j but the shifted process ."/ .j."/ .1/ C t / is the standard regenerative process considered above. The renewal type relation (3.2.19) takes in this case a form of the renewal type relation (4.2.2). In this case again, A D fj g, and the probability Pij."/ .t / plays the role of the probability PQ ."/ .t; A/. The distribution function 0 Gij."/ .t / is regarded as the distribution function FQ ."/ .t / that generates this renewal type relation, and the prob."/ ."/ ."/ ability jf D 1  0f D 1  0 G .1/ replaces the stopping probability fQ."/ i0

ij

ij

in the transition period. Conditions A2 , T1 , E02 , and A3 imply, by Lemma 4.2.1, that ."/ .0/ (u) jfi0 ! jfi0 D 0 as " ! 0. Thus, condition D17 holds with the limit constant fQ.0/ D 0. ."/ The distribution function G .t / replaces the distribution function FOQ ."/ .t /. Conij

ditions A2 , T1 , E02 and M11 imply, by Lemma 4.2.4, that (v) limT !1 lim"!0 .1  ."/ ."/ Gij .T //  limT !1 lim"!0 Mij =T D 0. Thus, condition D18 holds. Finally, by applying the asymptotic relation (3.2.22) given in Theorem 3.2.5, we get the asymptotic relation (4.4.8) for the case where i ¤ j .

4.4.3 Mixed ergodic and limit theorems for perturbed semi-Markov processes with non-recurrent-without absorption states Let us formulate the basic results on transition phenomena for perturbed semi-Markov processes in the case where condition E002 holds. In this case, define

r.0/

.0/

D

8 ˆ
t g; t  0; i ¤ 0: Pi .t / D r ¤0

First, consider the case where conditions E02 , A3 hold. Also assume that condition B6 holds for some function 0  t ."/ ! 1 as " ! 0, i.e., f ."/ t ."/ !  as " ! 0. In this case, condition B6 also holds for the function t t ."/ for every t > 0 with the limit t . Theorem 4.4.6. Let conditions A2 , T1 , A3 , E02 , M11 hold, and 0  t ."/ ! 1 as " ! 0 such that condition B6 holds. Then ."/ Pi."/ .t t ."/ / D Pi f."/ g ! e t=m 0 > tt

.0/

as " ! 0;

(4.4.40)

t > 0:

Proof. Relation (4.4.40) is a direct corollary of asymptotic relation (4.4.8) given in Theorem 4.4.1. It can be obtained by using the function t ."/ D t t ."/ in relation (4.4.8) .0/ and summing up the pre-limit and the limit expressions over j 2 X1 in the left and the right-hand sides of (4.4.8). ."/

Relation (4.4.40) shows that the properly normalised absorption time 0 =t ."/ has an exponential limit distribution with the parameter =m.0/ . Marginal asymptotic relations (4.4.26) and (4.4.40) give a clear interpretation to the mixed asymptotic relation (4.4.8) in Theorem 4.4.1. This theorem is a mixed ergodic and limit theorem for perturbed semi-Markov processes and absorption times. Let us now consider the case where conditions E002 , A3 hold. Let us also assume that condition B7 holds for some function 0  t ."/ ! 1 as " ! 0, i.e., fO."/ t ."/ !  as " ! 0. In this case, condition B7 also holds for the function t t ."/ for every t > 0 with the limit t .

216

4 Perturbed semi-Markov processes

Theorem 4.4.7. Let conditions A2 , T1 , A3 , E002 , M11 hold, and 0  t ."/ ! 1 as " ! 0 such that condition B7 holds. Then ."/ g Pi."/ .t t ."/ / D Pi f."/ 0 > tt .0/

! .1  X1fi0 /e t=m

.0/

as " ! 0; t > 0:

(4.4.41)

Proof. Relation (4.4.41) is a direct corollary of the asymptotic relation (4.4.15) given in Theorem 4.4.2. It can be obtained by using in (4.4.15) the function t t ."/ instead of t ."/ and summing up over r ¤ 0 the pre-limit and the limit expressions in the left and the right-hand sides of (4.4.15).

4.5 Asymptotic solidarity properties for hitting times In this section, we will study asymptotic solidarity properties for moment generating functions of hitting times.

4.5.1 Moment generating functions for hitting times Let us introduce the moment generating functions, for  2 R1 , i ¤ 0; j 2 X, Z 1 Z 1 ."/ ."/ ."/ ."/ ."/ s ."/ e Fij .ds/; pij ./ D e s Qij .ds/ D ij ./pij ; ij ./ D 0

and, for  2 R1 ; i; j ¤ 0, ."/

0 ij ./ D

0

Z

1 0

."/

."/

."/

e s 0 Gij .ds/ D Ei e j .j

."/

< 0 /:

."/ ."/ ./ D 0 if pij D 0. Note that, by the definition, pij By the definition, these moment generating functions are non-negative and nondecreasing of the argument  2 R1 taking values in the interval Œ0; C1. However, it is obvious that these functions take finite values at least on the interval .1; 0. Moreover, these generating functions are continuous in points  at which they take finite values. The following characteristic equation plays an important role in the sequel: Z 1 ."/ ."/ e s 0 Gjj .ds/ D 1: (4.5.1) 0 jj ./ D 0

The following system of linear equations can be written for every  2 R1 ; i; j ¤ 0, by using the Markov property of the semi-Markov process ."/ .t / at the moment of the first jump and then calculating the corresponding expectation that depends on the position of this process at the moment of the first jump: X ."/ ."/ ."/ ."/ pik ./ 0 kj ./; i ¤ 0: (4.5.2) 0 ij ./ D pij ./ C k¤j;0

217

4.5 Asymptotic solidarity properties for hitting times

Relations that constitute system (4.5.2) hold for every  2 R1 , i; j ¤ 0, even in ."/ ./ take the value the cases where the moment generating functions 0 ij."/ ./ or pik ."/

."/ ."/ ij ./pij and ."/ pik D 0, respec-

C1. In these cases, one needs to take the products pij ./ D ."/

."/

."/

."/

."/

."/

pik ./0 kj ./ D ik ./pik 0 kj ./ as zeros if pij D 0 or tively. The system (4.5.2) can be rewritten in the following matrix form: ."/ 0 ¥j ./

2

where ."/ 0 ¥j ./

and

6 D6 4

."/

."/

D pj ./ C j P."/ ./ 0 ¥j ./;

."/ 0 1j ./

:: :

."/ 0 Nj ./

3

(4.5.3)

3 ."/ ./ p1j 7 6 ."/ :: 7; pj ./ D 6 : 5 4 ."/ pNj ./ 2

7 7; 5

2

3 ."/ ."/ ."/ ."/ p11 ./ : : : p1.j 1/ ./ 0 p1.j C1/ ./ : : : p1N ./ 6 7 :: :: :: :: :: ."/ 6 7: j P ./ D 4 : : : : : 5 ."/ ."/ ."/ ."/ pN1 ./ : : : pN.j 1/ ./ 0 pN.j C1/ ./ : : : pN N ./ ."/

."/

Note that we admit the case where some elements of the vectors pj ./, 0 ¥j ./, and the matrix j P."/ ./ can be equal to C1. One should also set the product 01 to be 0 when calculating the products that represent elements of these vectors and matrices and the elements of the vectors in the right-hand side of equality (4.5.3). The n-the power of this matrix j P."/ ./ is given by the following formula: ."/

."/

.j P."/ .//n D k Ei e n .j

."/

^ 0 > n; ."/ n D k/ k :

(4.5.4)

This equality is also valid and becomes C1 D C1 for elements of the matrices in the left and the right-hand sides if the corresponding expectation takes the infinite value. Also note that one should take the corresponding expectation to be zero if ."/ ."/ ."/ Pi fj ^ 0 > n; n D kg D 0. Let us now consider the matrix series jA

."/

./ D I C j P."/ ./ C .j P."/ .//2 C    :

(4.5.5)

Note that we admit for some elements of the matrix j A./ to take the value C1 in the case where the corresponding series in the right-hand side diverges, which also includes the case where one of the summands in this series equals to C1. Let us introduce, for every  2 R1 ; j; k ¤ 0, the random variables ."/

ıj k ./ D

1 X nD1

e 

."/ .n1/

."/

.j

."/

."/

^ 0 > n  1; n1 D k/:

218

4 Perturbed semi-Markov processes ."/

The random variable ıj k ./ may be improper, i.e., take the value C1 with a positive probability. In any case, by the Lebesgue theorem, ."/

Ei ıj k ./ D

1 X

Ei e 

."/ .n1/

."/

.j

."/

."/

^ 0 > n  1; n1 D k/:

(4.5.6)

nD1

This identity remains true and takes the form C1 D C1 in the case where at least one of the expectations under the summation sign takes the value C1 or the corresponding series diverges. Also note that one should take the expectations under ."/ ."/ ."/ the summation sign to be zero if Pi fj ^ 0 > n  1; n1 D kg D 0. Comparing (4.5.4) and (4.5.6) yields a formula that gives a probabilistic interpretation to elements of the matrix j A."/ ./, jA

."/

./ D I C j P."/ ./ C .j P."/ .//2 C    D k Ei ıj."/ ./ k: k

(4.5.7)

Identity (4.5.7) is also valid if the matrix series in the left hand-side diverges in the sense that at least one of the series that represents elements of this matrix diverges. This includes the case where at least one element of the corresponding series is equal to C1 or the expectation that represents the corresponding element of the matrix in the right-hand side equals C1. In this case, the equalities of the corresponding elements of these matrices just take the form C1 D C1. In the lemmas formulated below and their proofs, we adopt similar conventions, i.e., corresponding equalities or inequalities for components of the vectors can take the form C1 D C1 or C1  C1. Also, the product 1  0 is set to be 0. ."/

Lemma 4.5.1. Let condition E1 hold for the Markov chain n , n D 0; 1; : : :, and j 2 X1."/ . Then the following claims hold true for any given value of  2 R1 : .˛/ ."/ ."/ ."/ ."/ 0 ¥j ./ D j A ./pj ./; .ˇ/ 0 ¥j ./ is a minimal non-negative solution of the ."/

system of linear equations (4.5.3); ./ 0 ¥j ./ is a vector with all positive com."/

."/

ponents; .ı/ the vector 0 ¥j ./ is finite if and only if the vector pj ./ and the matrices j P."/ ./ and j A."/ ./ are finite; ./ if the matrix j P."/ ./ is finite then the inverse matrix ŒI  j P."/ ./1 exists if and only if j A."/ ./ is finite and, in this case, ."/ ."/ ."/ ."/ 1 j A ./ D ŒI  j P ./ ; ."/ 0 ¥j ./ is finite if and only if the vector pj ./ and the matrix j P."/ ./ are finite, and the inverse matrix ŒI  j P."/ ./1 exists, moreover, ."/ in this case, 0 ¥j ./ is a unique solution of the system of linear equations (4.5.3); ./ ."/

."/

if the vector 0 ¥j ./ is finite and the vector pj .0 / and the matrix j P.0 / are finite 00 for some 0 >  then there exists  < 00  0 such that the vector 0 ¥."/ j . / is finite.

219

4.5 Asymptotic solidarity properties for hitting times

Proof. Direct calculations using relation (4.5.6) yield the following formula: ."/ 0 ij ./

."/

."/

D Ei e j .j D Ei

1 X X

."/

< 0 / ."/

."/ ."/ e n1 .."/ n1 D k; j ^ 0 > n  1/

nD1 k¤0 ."/

 e  n .."/ n D j/ D

1 X X

."/

."/

."/

Ei e n1 .n1 D k; j

."/

."/

^ 0 > n  1/pkj ./

nD1 k¤0

D

X

."/

pkj ./

D

."/

."/

."/

Ei e n1 .n1 D k; j

."/

^ 0 > n  1/

nD1

k¤0

X

1 X

."/ pkj ./Ei ıj."/ ./;  2 R1 ; i; j ¤ 0: k

(4.5.8)

k¤0

Note that identities in (4.5.8) also hold if the corresponding series diverge, which includes the case where at least one expectation representing the corresponding summand equals C1. In this case, the equalities take the form C1 D C1. One also ."/ ."/ should set the expectations under the summation sign to zero if Pi fj ^ 0 > ."/

n  1; n1 D kg D 0; the product 1  0 is also equal to 0. Relations (4.5.7) and (4.5.8) yield the formula given in the proposition .˛/. By the definition, 0 ¥."/ j ./ is a vector with components taking values in the interval ."/

Œ0; 1. As was pointed above, the vector 0 ¥j ./ is a solution of system (4.5.3). Let v./ be a vector components of which take values in the interval Œ0; 1 and ."/ such that it is also a solution of the system (a) v./ D p."/ j ./ Cj P ./v./. Iterat."/

ing relation (a) yields that v./ is also a solution of the system (b) v./ D pj ./ C ."/

."/

pj ./j P."/ ./ C    C pj ./ .j P."/ .//n C .j P."/ .//nC1 v./ for every n  1 and,

."/ ."/ ."/ ."/ n thus, (c) v./  p."/ j ./ C pj ./j P ./ C    C pj ./ .j P .// for n  1. By ."/

letting n tend to 1 in (c), we get the following inequality: (d) v./  pj ./ C ."/

."/

pj ./j P."/ ./ C    D pj ./j A."/ ./ D

."/ 0 ¥j ./

."/ 0 ¥j ./.

Inequality (d) proves that

is a minimal non-negative solution of system (4.5.3), i.e., the claim .ˇ/ is true. Since j is a recurrent-without-absorption state, (e) for any i ¤ 0 there exists a ."/ ."/ ."/ natural number nij such that Pi fj ^ 0 > nij ; nij C1 D j g > 0. Denote Cj D ."/

."/

."/

."/

fk ¤ 0 W pkj > 0g. Obviously (e) implies that Pi fj ^ 0 > nij ; nij C1 D j g D P P ."/ ."/ ."/ ."/ ."/ k2Cj Pi fj ^ 0 > nij ; nij D kgpkj > 0, and, therefore, (f) k2Cj Pi fj ^

."/ 0."/ > nij ; ."/ nij D kg > 0. Let also n D maxi¤0 nij . Since pkj ./ > 0 if

220

4 Perturbed semi-Markov processes

."/

."/

."/

."/

pkj > 0, relation (d) implies that (g) 0 ¥j ./  pj ./ C pj ./j P."/ ./ C    C ."/

pj ./ .j P."/ .//n , while (f) implies that the vector in the right-hand side of inequality (g) has all positive components, i.e., the claim ./ is also proved. Since the vector jP

."/ 0 ¥j ./

."/ ."/ 0 ¥j ./ D pj ./ C ."/ ."/ D pj ./ C pj ./j P."/ ./

satisfies the relation (h)

."/ ."/ 0 ¥j ./, it also satisfies the relation (i) 0 ¥j ./ ."/ n ."/ nC1 ¥."/ ./ for n C p."/ 0 j j ./.j P .// C .j P .//

."/ ./

C 

 1.

."/

Let us assume that 0 ¥j ./ is a finite vector, which, as was proved above, also has

."/ ./, in this case all positive components. Then (i) implies that all the quantities pik

."/ i; k ¤ 0, are finite, i.e., (j) the vector p."/ j ./ and the matrix j P ./ are finite. Also, ."/

by making n increase to 1 in relation (k) and taking into account that pj ./ C ."/

."/

pj ./j P."/ ./ C    C pj ./.j P."/ .//n ! that (l) .j

P."/ .//nC1

."/ 0 ¥j ./

."/ 0 ¥j ./

as n ! 1, and (i), we get

! 0 as n ! 1, where the last relations means that ."/

all components of the pre-limit vectors tend to 0. Since 0 ¥j ./ is a finite vector with all positive components, (m) implies that (n) .j P."/ .//nC1 ! 0 as n ! 1, where the last relations means that all components of the pre-limit matrices tend to 0. As is known, relation (n) holds if and only if the matrix series j A."/ ./ D I C ."/ ."/ 2 ."/ j P ./C j P .// C   converges in norms, i.e., the matrix j A ./ is finite. Indeed, ."/ m relation (n) implies that there exists m such that jj P .// j  Lm < 1. Let us also denote Km D max1r m1 j.j P."/ .//r j. Then (n) implies the estimates (o) jj P."/ .//n j  Km .Lm /Œn=m ; n  1. Estimates (o) implies convergence of the matrix ."/ series j A."/ ./ in norms. Conversely, if the vector pj ./ and the matrix j A."/ ./ are

finite, then the vector 0 ¥."/ j ./ is finite by the formula given in .˛/. This proves claim .ı/. The matrix j A."/ ./ defined in formula (4.5.5) satisfies the following relations: (p) ."/ ."/ ."/ 2 ."/ ."/ ."/ 2 j A ./ D I C j P ./ C j P .// C    D I C j P ./.I C j P ./ C j P .// C ."/ ."/    / D I C j P ./j A ./. Let us now assume that the matrix j P."/ ./ is finite. In this case, if j A."/ ./ is also finite, then relation (p) can be rewritten as (r) I D j A."/ ./  ŒI  j P."/ ./. Relation (r) means that there exists the inverse matrix ŒI  j P."/ ./1 D j A."/ ./. Conversely, if the matrix j P."/ ./ is finite and the inverse matrix ŒI  j P."/ ./1 exists, it, by the definition, satisfies the relation (s) I D ŒI  j P."/ ./ŒI  j P."/ ./1 that can be rewritten in an equivalent form, ŒI  j P."/ ./1 D I C j P."/ ./ŒI  j P."/ ./1 . An iteration of this relation implies that (t) ŒI  j P."/ ./1 D I C j P."/ ./ C    C .j P."/ .//n C.j P."/ .//nC1 ŒI  j P."/ ./1 ; n  1. Relation (t) implies that the matrix series I C j P."/ ./ C    converges in norm, and, therefore, (u) .j P."/ .//nC1 ŒI  ."/ 1 ! 0 as n ! 1, where the last relation means that all components of the j P ./ pre-limit matrices tend to 0. Relations (t) and (u) imply, in an obvious way, that (v) ŒI  j P."/ ./1 D I C j P."/ ./ C    D j A."/ ./.

221

4.5 Asymptotic solidarity properties for hitting times

Claim ."/ follows from the claims .˛/ and ./. ."/ It remains to prove the claim ./. Since the functions pik ./, i; k ¤ 0, (which ."/

."/

should be 0 if pik D 0) are non-decreasing in , we have that (w) pik ./ < 1, i; k ¤ 0, for  < 0 and, moreover, these function are continuous for  < 0 . As was proved ."/ 1 exists, above, if the vector 0 ¥."/ j ./ is finite then the inverse matrix ŒI  j P ./ and, therefore, (t) det.I  j P."/ .// ¤ 0. But det.I  j P."/ .// is a continuous function of elements of the matrix j P."/ ./. Thus, due to (w), we also have that (x) det.I  ."/ 00 00 0 ."/ 00 1 j P . // ¤ 0 for some  <  <  . Thus, the inverse matrix ŒI  j P . / ."/ 00 exists. Therefore, by ./, the vector 0 ¥j . / is finite. Remark 4.5.1. It follows from the proof of claim ./ in Lemma 4.5.1 that the parameter 00 depends on ". In fact, define 000 D sup.00 >  W det.I  j P."/ .00 // ¤ 0/. Then any value  < 00 < min.0 ; 000 / can be used in this proposition. Let us also consider the moment generating functions, Z 1 ."/ ."/  ./ D e s j Gi0 .ds/ j i0 0

."/

."/

."/

D Ei e 0 .0 < j /;

 2 R1 ;

i; j ¤ 0;

and ."/ ij."/ ./ D 0 ij."/ ./ C j i0 ./ Z 1 ."/ ."/ D e s Gij."/ .ds/ D Ei e .j ^0 / ; 0

 2 R1 ;

i; j ¤ 0:

By the definition, these moment generating functions are non-negative and nondecreasing in  2 R1 that takes values in the interval Œ0; C1. However, it is obvious that these functions take finite values at least in the interval .1; 0. Moreover, these generating functions are continuous in points  in which they take finite values. For every  2 R1 , i; j ¤ 0, by using the Markov property of the semi-Markov process ."/ .t / at the moment of the first jump and calculating the corresponding expectation conditioned by the position of this process at the moment of the first jump, we can write the following system of linear equations: X ."/ ."/ ."/ ."/ pik ./ 0 kj ./; i ¤ 0: (4.5.9) j i0 ./ D pi0 ./ C k¤j;0

Relations that constitute the system (4.5.9) are satisfied for every  2 R1 , i; j ¤ ."/ ."/ 0, even if the moment generating functions 0 i0 ./ or ik ./ take the value C1. ."/

In such a case, we take the products pi0 ./ D ."/ ."/ ."/ ./pik 0 kj ./ ik

."/ ."/ i0 ./pi0

."/

."/

and pi0 ./0 kj ./ D

."/ ."/ to be zero if pij D 0 or pik D 0, respectively.

222

4 Perturbed semi-Markov processes

System (4.5.9) can be rewritten in the following matrix form: ."/ 0 ¥j ./

where

2 ."/ j ¥0 ./

6 D4

."/ ."/ D p."/ 0 ./ C j P ./ 0 ¥j ./;

."/ j 10 ./

:: :

3 7 5;

(4.5.10)

3 ."/ p10 ./ 7 6 ."/ :: p0 ./ D 4 5: : 2

."/ j N 0 ./

."/

pN 0 ./ ."/

Note that we admit the case in which some elements of the the vectors p0 ./ and ."/ ¥ j 0 ./ could be equal to C1. The product 0  1 is taken to be 0 when calculating the products of elements of these vectors and matrices and the elements of the vectors in the right-hand side of (4.5.10). Let us also define a vector ¥."/ j ./, the components of which are moment generating ."/

functions ij ./, i ¤ 0, i.e., ."/ ."/ ¥."/ j ./ D 0 ¥j ./ C j ¥0 ./:

(4.5.11)

The following lemma supplements Lemma 4.5.1. ."/

Lemma 4.5.2. Let condition E1 hold for the Markov chain n , n D 0; 1; : : :, and ."/ let j 2 X1 . Then we have the following for any given value of  2 R1 : .˛/ ."/ ."/ ."/ ."/ j ¥0 ./ D j A ./p0 ./; .ˇ/ j ¥0 ./ is a minimal non-negative solution of the ."/ system of linear equations (4.5.10); ./ the vector j ¥."/ 0 ./ is finite if the vector p0 ./ and the matrices j P."/ ./ and j A."/ ./ D ŒI  j P."/ ./1 are finite and, in this case, ."/ j ¥0 ./ is a unique solution of the system of linear equations (4.5.10); .ı/ in this case, ¥."/ j ./ is also a finite vector. Proof. The proof of claims .˛/ and .ˇ/ in Lemma 4.5.2 are absolutely similar to the proofs of .˛/ and .ˇ/ in Lemma 4.5.1. Claim ./ is a direct corollary of .ı/ in Lemma 4.5.1 and .˛/ in Lemma 4.5.2. Claim .ı/ directly follows from (4.5.11) and ./ in Lemma 4.5.1 and ./ in Lemma 4.5.2.

4.5.2 Solidarity properties of moment generating functions of hitting times As above, we let X1."/ and X2."/ to be the classes of all recurrent-without-absorption and non-recurrent-without-absorption states i ¤ 0, respectively, for the Markov chain ."/ n , n D 0; 1; : : : . Let begin from the remark that solidarity properties for exponential moments of hitting times differ of solidarity properties for power moments.

4.5 Asymptotic solidarity properties for hitting times ."/

."/

223

."/

As is known if the cyclic power moment Ei .i /n .i < 0 / < 1 for some ."/ ."/ ."/ ."/ recurrent-without-absorption state i 2 X1 then Ej .j /n .j < 0 / < 1 for all ."/

states j 2 X1 . ."/ ."/ However, inequality 0 i i ./ < 1 for some i 2 X1 and  > 0 does not imply ."/ ."/ that 0 jj ./ < 1 for other states j 2 X1 , j ¤ i . The following simple example illustrates this. Let the set X D f0; 1; 2g and the ."/ ."/ transition probabilities pij 2 .0; 1/, i; j D 1; 2. In this case, the set X1 D f1; 2g. ."/

Let also the distribution functions Fij."/ .t / D 1  e ai t , t  0, where ai."/ > 0, i D 1; 2. ."/ Simple calculations show that the moment generating function 0 i i ./ < 1 if and only if  < a1."/ ^ a2."/ and case, ."/ 0 i i ./

D

a1."/ 

p ."/ ."/ i i ai C

."/

D

."/ aj ."/ aj 

a1

C

ai."/ 

."/ ."/

."/ aj ."/ aj 

."/

2

p ."/ jj

  aj

."/

a1

."/



  1

The inequality

p ."/ ."/ ij ai

 a."/ j

p C ."/ i i

  ai

."/ pjj < 1, where i; j D 1; 2, i ¤ j , and, in this

."/ ai

 1C

 

C 

p ."/ ."/ jj aj ."/

aj 

."/ p ."/ j i aj

."/

pij

."/

aj ."/ aj

aj."/



."/ pjj

1

."/

ai 

p ."/ : ."/ j i aj

(4.5.12)

."/

pjj < 1 can be rewritten in the equivalent form as  < ."/ ."/

."/ ."/

aj pj i . Let, for example, a1 p12 < a2 p21 . Then, as follows from the remarks ."/

."/

."/ ."/

."/ ."/

above, 0 11 ./ < 1 but 0 22 ./ D 1 for any a1 p12 <  < a2 p21 . This example clarifies specific forms of cyclic solidarity properties for the moment generating functions as they presented below. ."/

Lemma 4.5.3. Let (i) condition E1 hold for the Markov chain n , and (ii) there exists ."/ ."/ a state i 2 X1 such that 0 i i .ˇi / 2 .1; 1/ for some ˇi > 0. Then we have the ."/ following: .˛/ there exists 0 < ˇ  ˇi such that 0 jj .ˇ/ 2 .1; 1/ for every state ."/

."/

."/

j 2 X1 ; .ˇ/ 0 kj .ˇ/ < 1 for all states k; j 2 X1 ; ./ there exists a unique root ."/

."/

."/ of the equation 0 jj ./ D 1 for every state j 2 X1 ; .ı/ the root ."/ is the same ."/

for all states j 2 X1 ; ./ 0  ."/ < ˇ; ."/ if in addition to the assumptions (i) and

224

4 Perturbed semi-Markov processes ."/

."/

."/

(ii), the assumption (iii) 0 ki .ˇi / < 1, k 2 X2 , also holds, then ./ 0 kj .ˇ/ < 1, ."/

."/

j 2 X1 , k 2 X2 , for ˇ defined in .˛/. ."/

Proof. The lemma is trivial if the set X1 contains only one state. Thus, we assume that this set contains at least two states. Using the Markov property possessed by the semi-Markov process ."/ .t /, t  0, at the first hitting time, we can write the following renewal relations for two states i ¤ j , i; j ¤ 0: ."/ 0 Gi i .t /

."/ ."/ D 0j Gi."/ i .t / C 0i Gij .t /  0 Gj i .t /;

."/ 0 Gj i .t /

D 0j Gj i .t / C 0i Gjj .t /  0 Gj i .t /;

t  0;

(4.5.13)

t  0:

(4.5.14)

and ."/

."/

."/

These relation can be rewritten in terms of the corresponding moment generating functions for i ¤ j , i; j ¤ 0 in the following equivalent form: ."/ 0 i i ./

D

."/ 0j i i ./

."/ ."/ 0i ij ./ 0 j i ./;

 2 R1 ;

(4.5.15)

."/ 0 j i ./

D

."/ ."/ ."/ 0j j i ./ C 0i jj ./ 0 j i ./;

 2 R1 :

(4.5.16)

C

and

Similar two relations can be obtained by interchanging the states i and j in (4.5.15) and (4.5.16), ."/ 0 jj ./

D

."/ 0i jj ./

."/ ."/ 0j j i ./ 0 ij ./;

 2 R1 ;

(4.5.17)

."/ 0 ij ./

D

."/ ."/ ."/ 0i ij ./ C 0j i i ./ 0 ij ./;

 2 R1 :

(4.5.18)

C

and

Relations (4.5.15)–(4.5.18) can take the form C1 D C1 if the corresponding moment generating functions take the value C1. Here, we take 1  0 to be 0. Let us now assume that (a) i; j 2 X1."/ , i.e., both are recurrent-without-absorption states. This assumption imply that, for any  2 R1 , ."/ ."/ ."/ ."/ ."/ ."/ 0 i i ./; 0 jj ./; 0 ij ./; 0 j i ./; 0i ij ./; 0j j i ./

2 .0; 1:

(4.5.19)

Let us also assume that (b) 0 i."/ i ./ < 1 for some  > 0. Assumptions (a) and (b), and relations (4.5.15) and (4.5.19) imply that, for the value of  satisfying assumption (b), ."/ ."/ ."/ 0j i i ./; 0i ij ./; 0 j i ./

< 1:

(4.5.20)

225

4.5 Asymptotic solidarity properties for hitting times

Then, assumptions (a) and (b), and relations (4.5.16), (4.5.19), and (4.5.20) imply that if  satisfies the assumption (b), then ."/ ."/ 0j j i ./; 0i jj ./

< 1:

(4.5.21)

Moreover, relation (4.5.16) can be rewritten as 1 D ."/ 0i jj ./.

."/ ."/ 0j j i ./=0 j i ./

C

Therefore, the assumptions (a) and (b), and relations (4.5.16), (4.5.19), and (4.5.20) imply that ."/ 0i jj ./

(4.5.22)

< 1;

if  satisfies (b). However, the assumptions (a) and (b) and relations (4.5.16)–(4.5.22) do not guar."/ ./ and 0 ij."/ ./ are finite. antee that 0 jj

Let us now assume that (c) 0 i."/ i ./ 2 Œ0; 1 for some  > 0. Note that the assumption (c) is stronger than the assumption (b). In this case, relations (4.5.15), (4.5.19), and (4.5.20) imply that ."/ 0j i i ./

2 Œ0; 1/:

(4.5.23)

Let us again consider the return times j."/ .n/ D min.n > j."/ .n  1/ W ."/ n D j /,

n D 1; 2; : : :, j."/ .0/ D 0 and j."/ .n/ D ."/ .j."/ .n//, n D 0; 1; : : : . By the def-

inition, j."/ .n/ and j."/ .n/ are the times of the n-th return into the state j for the ."/ imbedded Markov chain ."/ n and for the semi-Markov process  .t /, respectively. A direct calculation yields the following formula for every  2 R1 : ."/ 0 ij ./

."/

."/

D Ei e j .j

."/

< 0 /

."/

D Ei e i .j."/ < 0."/ ^ i."/ / C Ei

1 X

."/

e .i

.n/Cj."/ ."/ i .n//

nD1

 D

.i."/ .n/

."/ 0i ij ./

C

1 X


0. ."/

First of all note that assumption (h) implies that (i) the distribution function 0 Gi i .t / ."/ is not concentrated in zero. Indeed, if this would be so, then the identity 0 i i ./ D ."/ 0 fi i  1,  2 R1 would hold. ."/ It follows from (h) and (i) that the moment generating function 0 i i ./ is a contin."/ ."/ uous and strictly increasing function for   ˇi . Note also that 0 i i .0/ D 0 fi i  1. ."/ ."/ Therefore, (j) there exists a unique root i of the equation 0 i i ./ D 1, and, moreover, (k) i."/ 2 Œ0; ˇi /. ."/

."/

Note that according to relation (4.5.15), 0j i i ./ < 0 i i ./ for any  in the ."/ ."/ assumption (c). Thus, 0j i i .i / < 1 and, therefore, the equality (4.5.27) can hold

4.5 Asymptotic solidarity properties for hitting times ."/

for  D i ."/ 0 jj ./ ."/

."/

."/

."/

if and only if 0 jj .i / D 1. Thus, (l) i

D 1 for every j 2

."/ X1 .

227

is also a root of the equation

By (l), we can omit the index i and denote this root

by  . All statements in Lemma 4.5.3 deal with the case where the initial state j belongs ."/ ."/ to the class of all recurrent-without-absorption states X1 for the Markov chain n . Since, by the definition, the states in set of non-recurrent-without-absorption states ."/ ."/ X2 can not be reached by the Markov chain n that starts in an arbitrary initial ."/ state j 2 X1 , we can, therefore, reduce the consideration to the case where the set ."/ X1 D f1; : : : ; N g, and this allows to apply Lemma 4.5.1. As was proved above (see, relation (4.5.20)), the assumption (h) implies that (m) ."/ ."/  0 ki .ˇi / < 1, k 2 X1 . Thus, we can apply Lemma 4.5.1. According to this

lemma, relations (m) imply that the vector p."/ i .ˇi / and the matrix i P.ˇi / are finite. In ."/ ."/ ."/ 0 fact, this means that (m ) pr k .ˇi / < 1, r; k 2 X1 . Thus, (m00 ) the vector pj .ˇi / and the matrix j P.ˇi / are finite. As was proved above (again see relation (4.5.20)), the ."/ ."/ ."/ relation 0 jj .."/ / D 1 implies that (n) 0 kj .."/ / < 1, k 2 X1 . Using the last statement of Lemma 4.5.1 we get that (o) relations (n) and (m00 ) imply that (p) there ."/ ."/ exist ."/ < ˇj < ˇi such that 0 kj .ˇj / < 1; k 2 X1 . ."/ Note also that the proposition (i) implies that (r) the distribution function 0 Gjj .t /

is not concentrated in zero for every j 2 X1."/ .

."/ It follows from (p) and (r) that the moment generating function 0 jj ./ is contin."/ uous and strictly increasing for   ˇj . Thus (s)  is the unique root of the equation ."/ ."/ ."/ 0 jj ./ D 1, and (t) 0 jj .ˇj / 2 .1; 1/. Let now ˇ D min.ˇj ; j 2 X1 /. By

.ˇ/ < 1 for r; k 2 X1."/ , and (v) ."/ < ˇ. Therefore, (w) the definition, (u) 0 r."/ k

."/ 0 jj .ˇ/

2 .1; 1/; for j 2 X1."/ .

Let us now go back and consider the general case where, in general, X2."/ ¤ ¿ and, ."/ in addition to (h), assume that (x) 0 ki .ˇi / < 1; k 2 X2."/ . For Z X, introduce the distribution functions ."/ Z Gij .t /

."/

."/

."/

D Pi fj  t; j

 Z g;

t  0;

i; j; k ¤ 0;

and the corresponding moment generating functions ."/ Z ij .t /

D Ei e

."/

j

.j."/



."/ Z /

Z D

0

1

e t Z Gij."/ .dt /;

 2 R1 ;

i; j ¤ 0:

Using the Markov property of the semi-Markov process ."/ .t /, t  0, at the first ."/ hitting time, we can write the following renewal relation for three states j 2 X1 ,

228

4 Perturbed semi-Markov processes ."/

."/

k 2 X2 , and the set Z ."/ D X1 [ f0g: ."/ 0 Gkj .t /

X

D

."/ Z ."/ Gkr .t /

."/  0 Grj .t /;

t  0:

(4.5.28)

."/ r 2X1

This relation can be rewritten in an equivalent form in terms of the corresponding moment generating functions, X ."/ ."/ ."/  2 R1 : (4.5.29) 0 kj ./ D Z ."/ kr ./0 rj ./; ."/

r 2X1

Relation (4.5.29) may take the form C1 D C1 if the corresponding moment generating functions take the value C1. In such a case, we set 1  0 to 0. ."/ Relations (4.5.19) and (u) imply that (y) 0 rj .ˇ/ 2 .0; 1/, r; j 2 X1."/ . Thus,

."/ .ˇ/ < 1, assumption (x) and relation (4.5.29), used for j D i , imply that (z) Z ."/ kr ."/

."/

."/

r 2 X1 , k 2 X2 . Finally, relations (y), (z), and (4.5.29) imply that 0 kj .ˇ/ < 1 ."/

."/

for any j 2 X1 , k 2 X2 . Remark 4.5.2. It follows from the proof of Lemma 4.5.3 that the parameter ˇ depends on ". In fact, define ˇj0 D sup.ˇj > ."/ W det.I  j P."/ .ˇj // ¤ 0/, j 2 X1."/ . Then

any values ."/ < ˇj < min.ˇj0 ; ˇi /, j 2 X1."/ can be used to define ˇ D min.ˇj ; j 2

X1."/ /.

The following lemma supplements Lemma 4.5.3. ."/

Lemma 4.5.4. Let (i) condition E1 hold for the Markov chain n ; (ii) there exist a ."/ ."/ ."/ state i 2 X1 such that 0 i i .ˇi / 2 .1; 1/ for some ˇi > 0; (iii) 0 ki .ˇi / < 1, ."/

."/

."/

."/

k 2 X2 ; and (iv) pk0 .ˇ/ < 1, k ¤ 0. Then .˛/ j k0 .ˇ/ < 1 and kj .ˇ/ < 1 ."/

for all states k ¤ 0, j 2 X1 , and ˇ defined in statement ./ of Lemma 4.5.3. Proof. As was shown in the proof of Lemma 4.5.3 (a), the assumptions (i)–(iii) imply ."/ that 0 kj .ˇ/ < 1 for all k ¤ 0, j 2 X1."/ . By Lemma 4.5.1, (a) implies that (b) the ."/

matrices j A."/ .ˇ/ D ŒI  j P."/ .ˇ/1 are finite for all j 2 X1 . Claim (b), assumption (iv) and Lemma 4.5.2 yield the statement of Lemma 4.5.4 in an obvious way.

4.5.3 Test functions and upper and lower bounds for exponential moments of hitting times Let us formulate simple necessary and sufficient conditions in terms of the so-called ."/ test functions for the relation 0 jj ./ 2 .1; 1/ to hold.

229

4.5 Asymptotic solidarity properties for hitting times ."/

Lemma 4.5.5. Let condition E1 hold for the Markov chain n , n D 0; 1; : : :, and j ¤ 0 be a recurrent-without-absorption state. Then the following takes place for any ."/ given  2 R1 : .˛/ 0 ¥j ./ is a finite vector if and only if there exists a finite vector ."/ v./ with non-negative components such that v./  p."/ j ./ C j P ./v./ and, in ."/

this case, 0 ¥j ./  v./; .ˇ/ if also there exists a vector u./  v./ with non."/

."/

negative components such that u./  pj ./ C j P."/ ./u./, then 0 ¥j ./  u./;

."/ ./ 2 .1; 1/ if and only if there exists a vector u./ described in .ˇ/ such ./ 0 jj that the j -th component of this vector is greater than 1.

Proof. Let v./ be a finite vector with non-negative components and satisfying the ."/ system of test inequalities, (a) v./  pj ./ Cj P."/ ./v./. Iterating the relation (a)

."/ ."/ ."/ ."/ n yields the inequalities (b) v./  p."/ j ./Cpj ./j P ./C  Cpj ./ .j P .// C ."/

."/

."/

.j P."/ .//nC1 v./  pj ./ C pj ./j P."/ ./ C    C pj ./ .j P."/ .//n for n  1. By letting n tend to 1 in (b), we get the following inequality: (c) v./  ."/ ."/ ."/ ."/ ."/ p."/ j ./ C pj ./j P ./ C    D pj ./j A ./ D 0 ¥j ./. Inequality (c) proves ."/

that 0 ¥j ./ is a finite vector if there exist a finite vector with non-negative components v./ that satisfies the system of inequalities (a). Moreover, in this case, the inequality 0 ¥."/ j ./  v./ also holds. ."/

On the other hand, the vector 0 ¥j ./, by the definition, has non-negative components and satisfies the system of linear equations (4.5.3), which is a particular case of the system (a). Thus, in the case where 0 ¥."/ j ./ is a finite vector, it can serve as the vector v./ satisfying system (a). Let u./ be a vector with non-negative components such that u./  v./ and sat."/ isfying the system of test inequalities (d) u./  pj ./ C j P."/ ./u./. By iterating ."/

."/

the relation (d), we get yields inequalities (e) u./  pj ./ C pj ./j P."/ ./ C    C

."/ n ."/ nC1 u./ for n  1. Since ¥."/ ./ is a finite vector, usp."/ 0 j j ./ .j P .// C.j P .// ing what was proved in Lemma 4.5.1 we get (f) .j P."/ .//n ! 0 as n ! 0, where the last relations means that all components of the pre-limit vectors tend to 0. Relation (f) implies that (g) .j P."/ .//nC1 u./ ! 0 as n ! 0, where the last relation means that all components of the pre-limit vectors tend to 0. By making n increase to 1 in (e) and ."/ ."/ ."/ using (g) we get the inequalities (h) u./  pj ./Cpj ./j P."/ ./C   D 0 ¥j ./. If the j -th component of the vector u./ is greater than 1, then the j -th component of ."/ the vector 0 ¥j ./ is also greater than 1.

On the other hand, the vector 0 ¥."/ j ./, by the definition, has non-negative components, and it satisfies the system of linear equations (4.5.3) that are also a particular ."/ case of system (d). Thus, in the case where 0 ¥j ./ is a finite vector, we can take it as the vector u./ that satisfies system (d).

230

4 Perturbed semi-Markov processes

4.5.4 Asymptotic Cram´er type conditions for hitting times In what follows, we assume that conditions A2 , T1 and E2 hold. .0/ .0/ Consider the sets X1 and X2 of recurrent-without-absorption and non-recurrent.0/ without-absorption states for the limiting Markov chain n , respectively. Recall that, according to condition E2 , X1.0/ ¤ ¿, while X2.0/ D ¿ if condition E02 holds and X2.0/ ¤ ¿ if condition E002 holds. We will assume the following Cram´er type condition: ."/

C12 : There exists ı > 0 such that lim0"!0 pij .ı/ < 1, i ¤ 0, j 2 X. The following condition obviously implies condition C12 to hold: C012 : There exists ı > 0 such that lim0"!0

."/ ij .ı/

< 1, i ¤ 0, j 2 X.

Condition C12 implies that there exists "01 > 0 such that .0/ X1

."/ ij .ı/

< 1, i ¤ 0,

"01 .

for "  j 2 The following condition plays a key role in our considerations: .0/

C13 : There exist i 2 X1

and ˇi 2 .0; ı, for ı defined in condition C12 , such that

.0/

(a) 0 i i .ˇi / 2 .1; 1/; .0/

.0/

(b) 0 ki .ˇi / < 1; k 2 X2 . Note that condition C13 (b) is automatically satisfied if condition E02 holds, i.e., if .0/ X2 D ¿. Lemma 4.5.6. Let conditions A2 , T1 , E2 , C12 , and C13 hold. Then, .˛/ there exist ."/ .ˇ/ 2 .1; 1/ for every 0 < ˇ  ˇi and "00 > 0 (defined by (4.5.30)) such that 0 jj ."/ .ˇ/ < 1 for all states k ¤ 0, j 2 X1.0/ , and state j 2 X1.0/ and "  "00 ; .ˇ/ 0 kj

."/ ./ D 1 for every "  "00 ; ./ there exists a unique root ."/ of the equation 0 jj

state j 2 X1.0/ and "  "00 ; .ı/ the root ."/ is the same for all states j 2 X1.0/ ; ./ ."/ .0/ 0  ."/ < ˇ for "  "00 ; ."/ ."/ ! .0/ as " ! 0; ./ 0 kj ./ ! 0 kj ./ as

" ! 0 for all k ¤ 0, j 2 X1.0/ , and   ˇ.

Proof. Let us first apply Lemma 4.5.3 to the case where " D 0. According to condition .0/ C13 , this lemma implies that (a) there exists 0 < ˇ  ˇi such that 0 jj .ˇ/ 2 .1; 1/, .0/

.0/

.0/

j 2 X1 ; (a0 ) 0 kj .ˇ/ < 1 for k ¤ 0, j 2 X1 ; (a00 ) there exists a unique root .0/ .0/

.0/

of the equation 0 jj ./ D 1 for every state j 2 X1 ; (a000 ) the root .0/ is the same .0/

for all states j 2 X1 ; (a0000 ) 0  .0/ < ˇ.

4.5 Asymptotic solidarity properties for hitting times

231

Lemma 4.5.1 and (a0 ) implies that, in this case, (b) there exists the inverse matrix ŒI  j P."/ .ˇ/1 for any j 2 X1.0/ , i.e., (b0 ) det.I  j P.0/ .ˇ// ¤ 0 for j 2 X1.0/ . Conditions T1 and C12 imply that (c) the moment generating functions converge, ."/ .0/ pkr ./ ! pkr ./ as " ! 0 for k; r ¤ 0, and   ˇ. Since the determinant of a matrix is a continuous function of elements of the matrix, relations (b0 ) and (c) imply that there exists "02 > 0 such that (d) det.I  j P."/ .ˇ// ¤ 0 .0/ ."/ ."/ for j 2 X1 and "  min."01 ; "02 /. Moreover, (e) 0 ¥j .ˇ/ D ŒI  j P."/ .ˇ/1 pj .ˇ/, j 2 X1.0/ . Since, the elements of the inverse matrix are continuous functions of elements of the matrix itself, (c) implies that (c0 ) ŒI  j P."/ .ˇ/1 ! ŒI  j P.0/ .ˇ/1 as " ! 0, ."/ .0/ j 2 X1.0/ . Now, (c), (c0 ), and (e) imply that (f) 0 kj .ˇ/ ! 0 kj .ˇ/ < 1 as " ! 0, .0/

for k ¤ 0, j 2 X1 . Relations (a0 ) and (f) imply that there exists "3 > 0 such that ."/ ."/ (g) 0 jj .ˇ/ 2 .1; 1/ for j 2 X1.0/ and "  "00 and (h) 0 kj .ˇ/ < 1 for k ¤ 0, .0/

j 2 X1 and "  "00 , where "00 D min."01 ; "02 ; "03 /:

(4.5.30)

Lemma 4.5.3 and relations (g) and (h) now give statements .˛/–./ of Lemma 4.5.6. ."/ ./ ) According to Lemma 4.2.3, conditions A2 , T1 , and E2 imply that (i) 0 Gkj .0/ 0 Gkj ./ ."/ 0 kj ./

.0/

as " ! 0, k ¤ 0, j 2 X1 . Relations (f) and (i) easily give that (f0 ) .0/

.0/

! 0 kj ./ as " ! 0 for all k ¤ 0, j 2 X1 , and   ˇ, so that we see that claim ./ holds. .0/ Note that relation (a) implies that (j) the distribution function 0 Gjj .t / is not con.0/

centrated in zero for every j 2 X1 . Finally, relations (i) and (f0 ), both taken for k D j , and proposition (j) yield, by Lemma 1.4.3, that statement ."/ of Lemma 4.5.6 holds. The following lemma supplements Lemma 4.5.6. Lemma 4.5.7. Let conditions A2 , T1 , E2 , C12 , and C13 hold. Then, .˛/ there exists ."/ ."/ "04 > 0 (defined in formula (4.5.31)) such that j k0 .ˇ/ < 1 and kj .ˇ/ < 1 for .0/

0 < ˇ  ı defined in .˛/ of Lemma 4.5.6, k ¤ 0, j 2 X1 , and "  "04 ; .ˇ/ ."/ .0/ ."/ .0/ j k0 ./ ! j k0 ./ < 1 as " ! 0 and kj ./ ! kj ./ < 1 as " ! 0 for .0/

  ˇ and k ¤ 0, j 2 X1 . Proof. As was shown in the proof of Lemma 4.5.6, (k) the inverse matrices ŒI  ."/ 1 exist for j 2 X .0/ and "  "0 , and (k0 ) ŒI P."/ .ˇ/1 ! ŒI P.0/ .ˇ/1 j P .ˇ/ j j 0 1 .0/ as " ! 0, j 2 X1 .

232

4 Perturbed semi-Markov processes ."/

Condition C12 implies that there exists "05 > 0 such that (l) pk0 .ˇ/ < 1, k ¤ ."/

0, for "  "05 . According to Lemma 4.5.2, (k) and (l) imply that (m) j ¥0 .ˇ/ D ."/ .0/ ŒI  j P."/ .ˇ/1 p0 .ˇ/, for j 2 X1 and "  "04 , where "04 D min."00 ; "05 /: ."/

(4.5.31) .0/

Conditions T1 (b) and C12 imply that (n) pk0 ./ ! pk0 ./ as " ! 0 for k ¤ 0 ."/

.0/

and   ˇ. Relations (k0 ), (m), and (n0 ) imply that (o) j k0 .ˇ/ ! j k0 .ˇ/ < 1 as .0/

" ! 0 for k ¤ 0, j 2 X1 . ."/ ./ ) According to Lemma 4.2.3, conditions A2 , T1 , and E2 imply that (p) j Gk0

.0/ j Gk0 ./ ."/ j k0 ./

as " ! 0, k ¤ 0, j 2 X1.0/ . Relations (o) and (p) easily give that (o0 ) !

.0/ j k0 ./

as " ! 0 for k ¤ 0, j 2 X1.0/ and   ˇ. Finally, (o0 ) and

."/ .0/ statement ./ of Lemma 4.5.6 imply that (o00 ) kj ./ ! kj ./ as " ! 0 for k ¤ 0, .0/

j 2 X1 and   ˇ. Remark 4.5.3. It follows from the proof of Lemmas 4.5.6 and 4.5.7 that the assump."/ tion lim0"!0 pi0 .ı/ < 1, i ¤ 0, is not needed in Lemma 4.5.6, but it is required in Lemma 4.5.7.

4.5.5 Sufficient asymptotic Cram´er type conditions for hitting times Let us recall formula (4.2.29), .0/

.0/

1  0 Gjj .1/ D jfj 0 D

N X

.0/ .0/

Ej ıj k pk0 ;

.0/

j 2 X1 :

(4.5.32)

kD1

It follows from formula (4.5.32) that, (a) under conditions A2 and E2 , the distribu.0/ .0/ tion functions 0 Gjj .t / are proper or improper simultaneously for all j 2 X1 , and that (b) condition A3 is necessary and sufficient for these distribution functions to be proper. .0/ Also recall that, (c) under conditions A2 and E2 , the distribution functions 0 Gjj .t / .0/

are concentrated or not concentrated in zero simultaneously for all j 2 X1 , and that (d) condition I4 (a) is necessary and sufficient for these distribution functions to be not concentrated in zero. .0/ .0/ If 0 Gjj .t / is a proper distribution function and not concentrated, then 0 jj .0/ D

.0/ .0/ ./ 2 .0; 1/ for  < 0, and 0 jj ./ 2 .1; 1 for any  > 0. Therefore, 1, while 0 jj (e) 0 in this case, is a unique root for the characteristic equation (4.5.1). .0/ If 0 Gjj .t / is an improper distribution function, i.e., condition A2 does not hold .0/

.0/

and 0 jj .ˇ/ < 1 for some ˇ > 0 and 0 Gjj .t / is not concentrated in zero, then

233

4.5 Asymptotic solidarity properties for hitting times .0/ 0 jj ./ is a continuous and strictly increasing function for   .0/ .0/ .0/ case, 0 jj .0/ D 0 Gjj .1/ < 1. Thus, (f) if 0 jj .ˇ/ 2 .1; 1/ .0/

ˇ. Moreover, in this there exists a unique

root 0 <  < ˇ of the characteristic equation (4.5.1). The following lemma gives an additional useful information about the characteristic roots ."/ . Lemma 4.5.8. Let conditions A2 , T1 , I4 , E2 , C12 , and A3 hold. Then, .˛/ condition .0/ C13 holds for any i 2 X1 and 0 < ˇi  ˇ .0/ for some 0 < ˇ.0/  ı; .ˇ/ ."/ ! .0/  D 0 as " ! 0. Proof. As was shown in the proof of Lemma 4.2.1, (g) conditions A2 and E2 imply that .0/ the inverse matrix ŒI  i P.0/ 1 exists for every i 2 X1 , i.e., (h) det.I  i P.0/ / ¤ 0. .0/ .0/ .0/ By condition C12 , (i) pkr ./ ! pkr .0/ D pkr as  ! 0 for all k; r ¤ 0, and thus (j) i P.0/ ./ ! i P.0/ as  ! 0. Relations (h) and (j) imply that (k) there exists small .0/ enough 0 < ˇ.0/  ı such that det.I  i P.0/ .ˇ .0/ // ¤ 0 for every i 2 X1 . Also, .0/ .0/ .0/ by condition C12 , (l) pkr .ˇ / < 1, k; r 2 X1 . Now, Lemma 4.5.1, (k) and (l) .0/

.0/

.0/

imply that (m) 0 i i .ˇ .0/ / < 1, i 2 X1 . Note that (n) 0 i i ./ > 1 for  > 0 .0/ since, due to conditions I4 and A3 , the distribution function 0 Gi i .t / is proper and not concentrated in zero. Relations (m) and (n) prove .˛/. The statement .ˇ/ follows from Lemma 4.5.6 and the remarks above which imply that (o) .0/ D 0. Let us introduce moment generating functions X ."/ pi."/ ./ D pij ./;   0;

i ¤ 0;

j ¤0

and define for i ¤ 0, .0/

i D sup.  0 W pi ./ < 1/: .0/

(4.5.33) .0/

By the definition, (z0 ) i 2 Œ0; 1; (z00 ) pi ./ < 1 for  < i and pi .i /  1; (z000 ) pi.0/ ./ ! pi.0/ .i / as  < i ;  ! i . Let us also define  D min i : .0/

i2X1

Let us now introduce the following condition: G1 :

(a)  > 0; .0/

.0/

(b) pi . / D 1 for some i 2 X1 .

(4.5.34)

234

4 Perturbed semi-Markov processes .0/

Lemma 4.5.9. Let conditions A2 , E2 , and G1 hold. Then, .˛/ there exist i 2 X1 .0/ and 0 < ˇi <  such that 0 i i .ˇi / 2 .1; 1/. .0/

Proof. Let i 2 X1 be a state such that condition G1 (b) holds. Then, (p) there .0/ .0/  exists k 2 X1.0/ such that pik > 0 and pik . / D 1. This obviously imply that .0/  0 i i . /

D 1. By repeating the first part of the proof of Lemma 4.5.8 we get that (r) there exists 0 < N <  such that det.I  i P.0/ .Ni // ¤ 0. Since 0 < Ni <  .0/ .0/ we also have (s ) pkr .Ni / < 1, k; r 2 X1 . Relations (r) and (s) imply that (t) .0/     0 i i .Ni / < 1. Relations (p) and (t) imply that (u) there exist Ni < Oi   such .0/ .0/ that 0 i i ./ < 1 for  < Oi and 0 i i ./ ! 1 as  < Oi ;  ! Oi . Since function .0/  0 i i ./ is continuous and strictly monotone for  < Oi , relations (u) imply that there .0/ exist Ni  ˇi < Oi such that 0 i i .ˇi / 2 .1; 1/.

Let us also introduce condition: ."/ ./ < 1 for  <  , i ¤ 0, j 2 X; G2 : (a): lim0"!0 pij .0/ (b): 0 ki ./ < 1 for  <  ; k 2 X2.0/ for some i 2 X1.0/ . .0/

Lemma 4.5.10. Let conditions A2 , E2 , G1 and G2 hold. Then, .˛/ there exist i 2 X1 and 0 < ˇi < ı <  such that conditions C12 and C13 hold.

."/ Proof. By condition G2 (a) and the definition of  , (v) pij ./ < 1 for i ¤ 0, j 2 X. Also, as follows from Lemma 4.5.9, (w) conditions A2 , E2 , and G1 imply that .0/ .0/ there exist i 2 X1 and ˇi <  such that 0 i i .ˇi / 2 .1; 1/. Relations (v), (w), and condition G2 imply that conditions C12 and C13 holds with any 0 < ˇi < ı <  .

Remark 4.5.4. Lemmas 4.5.9 and 4.5.10 let one to replace conditions and C13 by ."/ conditions G1 and G2 in Lemmas 4.5.6 and 4.5.7. The assumption lim0"!0 pi0 ./ < 1,  <  , i ¤ 0, is not needed in the corresponding variant of Lemma 4.5.6 but it is not required in the corresponding variant of Lemma 4.5.7. In conclusion, we would like to note that two cases are characteristic for most applications. Either we have a pseudo-stationary model, where the root .0/ D 0 or the quasi-stationary model, where the root .0/ > 0 and condition E02 holds, i.e., the set of non-recurrent-without-absorption states X2.0/ is empty. In the first case, Lemma 4.5.8 let one escape checking of the “unpleasant” condition C13 (b). In the second case, the “unpleasant” condition G2 , which is an analogue of C13 (b), can obviously be omitted in Lemma 4.5.10. This let one effectively use this lemma for checking condition C13 .

4.6 Mixed ergodic and large deviation theorems for semi-Markov processes

235

4.6 Mixed ergodic and large deviation theorems for perturbed semi-Markov processes In this section we formulate a so-called mixed ergodic and large deviation theorem for perturbed semi-Markov processes.

4.6.1 Pseudo-stationary exponential asymptotics for perturbed semiMarkov processes without non-recurrent-without-absorption states The following two theorems require Cram´er type conditions on transition times for semi-Markov processes but do not impose any restrictions on the rate of the time growth as it was in Theorems 4.4.1 and 4.4.2. .0/ Note that, in the theorem formulated below, the stationary probabilities j , j ¤ 0, are given by formula (4.4.7). Theorem 4.6.1. Let conditions A2 , T1 , E02 , C12 , A3 , and N1 hold. Then, for any 0  t ."/ ! 1 as " ! 0, ."/

Pi f."/ .t ."/ / D j; 0 > t ."/ g .0/ ! j as " ! 0; expf."/ t ."/ g

i; j ¤ 0:

(4.6.1)

Proof. The proof mainly repeats the proof of Theorem 4.4.1. According to the remarks made in Subsection 4.2.1, the semi-Markov process ."/ .t /, more precisely its modified version with the absorption state removed using the procedure described in that subsection, can be considered as a perturbed regenerative process. The absorption ."/ time 0 plays the role of the regenerative stopping moment. .0/

.0/

.0/

Let us chose some state j 2 X1 . Note that, by condition E02 , X1 D X0 D f1; : : : ; N g. We can apply Theorem 3.3.1 to prove the asymptotic relation (4.6.1) for the case where the initial state i D j , since ."/ .t / is a standard regenerative process with the regeneration times j."/ .n/ that are subsequent times of return of the process ."/ .t / into the state j . The renewal equation (3.2.1) takes now the form of the renewal equation (4.2.1). ."/ In this case, we have A D fj g and the probability Pjj .t / can be regarded as the probability P ."/ .t; A/. ."/ .t / replaces the distribution function F ."/ .t / that The distribution function 0 Gjj generates this renewal equation. As was pointed out in the proof of Theorem 4.4.1, conditions A2 , T1 , E02 , A3 , and N1 imply, by Lemmas 4.2.1, 4.2.3, 4.3.4, and 4.4.1, ."/ .0/ respectively, that (a) the distribution functions 0 Gjj .t / weakly converge to 0 Gjj .t / .0/

as " ! 0, (b) the limit distribution function 0 Gjj .t / is proper, and (c) it is also nonarithmetic. It follows from (a)–(c) that the condition of weak convergence D13 holds ."/ for the distribution functions 0 Gjj .t /.

236

4 Perturbed semi-Markov processes

."/ The distribution function Gjj .t / is regarded as the distribution function FO ."/ .t /. Conditions A2 , T1 , E02 , and C12 yield, by Lemmas 4.5.6, 4.5.7 and 4.5.8, (d) conver."/ .0/ .ˇ/ ! jj .ˇ/ < 1 as " ! 0. gence of the corresponding exponential moments, jj Lemmas 4.5.7 – 4.5.8 and the convergence relation (d) imply that condition C3 holds ."/ for the distribution functions Gjj .t /. ."/ The function 1Fj .t / is a forcing function for the renewal equation (4.2.1). These functions are (e) non-decreasing, and, (f) by condition T1 , weakly converge. As was ."/ shown in the proof of Theorem 4.4.1, (e) and (f) imply that the functions 1  Fj .t / converge locally uniformly in every point of the set Cj of positive continuity points of .0/ the limit function 1  Fj .t /. Note that the set Cj has the value of Lebesgue measure ."/

m.C j / D 0. Thus condition F7 holds for the functions 1  Fj .t /. All conditions of Theorem 3.3.1 hold. The corresponding evaluation of the station.0/ ary probabilities j has been made in the proof of Theorem 4.4.1. By applying the asymptotic relation (1.4.9), given in Theorem 3.3.1, we get the asymptotic relation (4.6.1) for i D j . We can apply Theorem 3.3.8 to prove the asymptotic relation (4.6.1) for the case where i ¤ j . In this case, ."/ .t / is again a regenerative process with regeneration ."/ times j .n/ that are subsequent times of return of the process ."/ .t / into the state j . This process has a transition period, since the initial state differs from j but the shifted ."/ process ."/ .j .1/ C t / is the standard regenerative process considered above. The renewal type relation (3.2.19) takes in this case the form of the renewal type ."/ relation (4.2.2). In this case again, A D fj g and the probability Pij .t / plays the role ."/ of the probability PQ ."/ .t; A/. The distribution function 0 Gij .t / can be regarded as the distribution function FQ ."/ .t / that generates this renewal type relation. ."/

."/

."/

The probability jfi0 D 1  0 fij D 1  0 Gij .1/ becomes the stopping probability fQ."/ in the transition period. Conditions A2 , T1 , E0 , and A3 imply, by Lemma 2

4.2.1, that (g) jfi0."/ ! jfi0.0/ D 0 as " ! 0. Thus, condition D17 holds with the limit constant fQ.0/ D 0. The distribution function Gij."/ .t / is now the distribution function FOQ ."/ .t /. Conditions A2 , T1 , E02 and C12 imply, by Lemmas 4.5.6, 4.5.7 and 4.5.8, that (h) condition C8 holds for these distribution functions. This condition implies conditions C7 and D25 . Finally, by applying the asymptotic relation (3.3.28) given in Theorem 3.3.8, we get the asymptotic relation (4.6.1) for the case where i ¤ j .

4.6.2 Pseudo-stationary exponential asymptotics for perturbed semiMarkov processes with non-recurrent-without-absorption states Let us now formulate a theorem which is an analogue of Theorem 4.6.1 for the case .0/ where the limit set of non-recurrent-without-absorption states is X2 ¤ 0.

237

4.6 Mixed ergodic and large deviation theorems for semi-Markov processes .0/

Note that the stationary probabilities j , j ¤ 0, and the absorption probabilities .0/

X1

.0/

fi0 are given in the theorem formulated below by formulas (4.4.28) and (4.2.77), .0/

respectively, with Z D X1 . Theorem 4.6.2. Let conditions A2 , T1 , E002 , C12 , A3 , and N1 hold. Then, for any 0  t ."/ ! 1 as " ! 0, ."/ g Pi f."/ .t ."/ / D r; ."/ 0 >t ! .1  X .0/ fi0.0/ /r.0/ as " ! 0; i; r ¤ 0: (4.6.2) 1 expf."/ t ."/ g .0/

Proof. Let us first consider that case where i D r and r D j 2 X1 . The first part of the proof in which conditions D13 , C3 , F7 are verified repeats the corresponding part in the proof of Theorem 4.6.1. We only need to replace the reference to the condition E02 with a reference to condition E002 . Thus again, all conditions of Theorem 3.3.1 hold true. An evaluation of the corre.0/ sponding stationary probabilities j was made in the proof of Theorem 4.4.1. There .0/

.0/

was also pointed out a proof that X .0/ fi0 D 0 for i 2 X1 . 1 By applying the asymptotic relation (1.4.9) given in Theorem 3.3.1, we get the asymptotic relation (4.6.1) for the case where i D r and j D r 2 X1.0/ . .0/

Let us now consider the case where the initial state i ¤ r; 0 and r D j 2 X1 . As above, the renewal type relation (3.2.19) takes the form of the renewal equation ."/ (4.2.2), the distribution function 0 Gij .t / becomes the distribution function FQ ."/ .t / ."/

."/

that generates this renewal type relation, and the probability jfi0 D 1  0 fij D ."/ 1  0 G .1/ plays the role of the stopping probability fQ."/ in the transition period. ij

."/

.0/

Conditions A2 , T1 , E002 , and A3 imply, by Lemma 4.2.1, that (a) jfi0 ! jfi0 as .0/ " ! 0. Thus, condition D17 holds with the limit constant fQ.0/ D jfi0 . As was .0/ .0/ shown in the proof of Theorem 4.4.2, jfi0 D X .0/ fi0 in this case. 1

Similarly to the above, the distribution function Gij."/ .t / can be regarded as the distribution function FOQ ."/ .t /. Conditions A2 , T1 , E02 and C12 imply, by Lemmas 4.5.6, 4.5.7 and 4.5.8, that (b) condition C8 holds for these distribution functions, which implies that conditions C7 and D25 are also verified. By applying the asymptotic relation (3.3.28) given in Theorem 3.3.8, we get the .0/ asymptotic relation (4.6.2) for the case where i ¤ r; 0 and r D j 2 X1 . .0/

It still remains to prove asymptotic relation (4.6.2) for r 2 X2 . In this case, we can .0/ choose some state j 2 X1 and use the renewal relations (4.2.3) and (4.2.6) instead of the renewal relations (4.2.1) and (4.2.2). .0/ .0/ Let us first consider the case where i D j 2 X1 ; r 2 X2 .

238

4 Perturbed semi-Markov processes

The renewal equations (4.2.3) and (4.2.1) have the same distribution function ."/ ."/ 0 Gjj .t / that generates these equations, and the distribution function Gjj .t / plays the role of the distribution function FO ."/ .t /. Thus, conditions D13 and C3 do not require a new proof. However, the renewal equation (4.2.3) has a forcing function that differs from the forcing function for the renewal equation (4.2.1). Condition F7 was checked in this .0/ case, in the proof of Theorem 4.4.2. The corresponding stationary probabilities r were also given in the proof of this theorem. By applying the asymptotic relation (1.4.9) given in Theorem 3.3.1, we get the asymptotic relation (4.6.2) for the case where i D j 2 X1.0/ , r 2 X2.0/ . The last case to be considered is the one where the initial state i ¤ j; 0, j 2 X1.0/ , .0/ and r 2 X2 . The renewal relation (3.2.19) takes now the form of the renewal relation (4.2.6) that should be used instead of the renewal relation (4.2.2). The renewal relations (4.2.6) ."/ and (4.2.2) have the same distribution function 0 Gij .t / generating these equations and ."/

the same distribution function Gij .t / that plays the role of the distribution function FOQ ."/ .t /. Thus, condition C8 and, consequently, conditions C7 and D25 do not need a new proof. Finally, by applying the asymptotic relation (3.3.28) given in Theorem 3.3.8, we get the asymptotic relation (4.6.2) for the case where i ¤ j; 0, j 2 X1.0/ , and r 2 X2.0/ . Remark 4.6.1. By Lemma 1.4.2, under conditions A2 , T1 , E2 C12 , A3 , we have ."/ 0 00 ."/  jfj."/ 0 =0 Mjj as " ! 0. Thus, if conditions E2 and B6 or E2 and B7 hold ."/ ."/ .0/ with a limit constant  > 0, then  t ! =m as " ! 0. In these cases, the asymptotic relations given in Theorems 4.6.1 and 4.6.2 reduce to the asymptotic relations given, respectively, in Theorems 4.4.1 and 4.4.2.

4.6.3 Quasi-stationary exponential asymptotics for perturbed semi-Markov processes without non-recurrent-without-absorption states Let us now consider the general case where the asymptotic non-absorption condition A3 may or may not hold. Note that the characteristic root is .0/ > 0 if and only if the condition A3 does not hold or, equivalently, .0/ D 0 if and only if condition A3 holds. Denote R 1 ."/ s .1  Fj."/ .s// ds 0 e ."/ ."/ ."/ ; j 2 X1 : Qj . / D R 1 (4.6.3) ."/ s ."/  se G .ds/ 0 jj 0 ."/

Note that, under conditions A2 , T1 , and E02 , X1 small enough. This follows from Lemma 4.2.2.

D X0 D f1; : : : ; N g for all "

4.6 Mixed ergodic and large deviation theorems for semi-Markov processes

239

Let us also denote ."/ ."/ ."/ QQ ij .."/ / D 0 ij .."/ /Qj .."/ /;

."/

i; j 2 X1 :

(4.6.4)

Note that, under conditions A2 , T1 , E02 , C12 , and C13 , the integrals in the expression ."/ ."/ that defines the quantities Qj .."/ / and QQ ij .."/ / converge for all " small enough. This follows from Lemma 4.5.7. Theorem 4.6.3. Let conditions A2 , T1 , E02 , C12 , C13 , and N1 hold. Then, for any 0  t ."/ ! 1 as " ! 0, ."/

Pi f."/ .t ."/ / D j; 0 > t ."/ g .0/ ! QQ ij ..0/ / as " ! 0; ."/ ."/ expf t g

i; j ¤ 0:

(4.6.5)

Proof. Using the remarks made in the Subsection 4.2.1 we see that the semi-Markov process ."/ .t /, more precisely its modified version with the absorption state removed according to the procedure described in that subsection, can be considered as a per."/ turbed regenerative process. The absorption time 0 plays the role of the regenerative stopping moment. .0/ .0/ Let us chose some state j 2 X1 . Note that, according to condition E02 , X1 D X0 D f1; : : : ; N g. In this case, we can apply Theorem 3.3.2 to prove the asymptotic relation (4.6.5) for the case where i D j , since in this case, ."/ .t / is a standard re."/ generative process with regeneration times j .n/ that are subsequent times of return of the process ."/ .t / into the state j . The renewal equation (3.2.1) takes now the form of renewal equation (4.2.1). In this ."/ case, A D fj g and the probability Pjj .t / is regarded as the probability P ."/ .t; A/. ."/

The distribution function 0 Gjj .t / is now the distribution function F ."/ .t / that generates this renewal equation. As was pointed out in the proof of Theorem 4.4.1, conditions A2 , T1 , E02 , A3 , and N1 imply, by Lemmas 4.2.1, 4.2.3, 4.3.4, and 4.4.1, respec."/ .0/ tively, that (a) the distribution functions 0 Gjj .t / weakly converge to 0 Gjj .t / as " ! ."/

.0/

.0/

0, (b) the quantities 0 Gjj .1/ converge to 0 Gjj .1/ as " ! 0, (c) 0 Gjj .1/ > 0 .0/

.0/

and the normalised limit distribution function 0 Gjj .t /=0 Gjj .1/ is non-arithmetic. It follows from (a)–(d) that the condition of weak convergence D22 holds for the dis."/ tribution functions 0 Gjj .t /. ."/ The distribution function Gjj .t / replaces the distribution function FO ."/ .t /. Conditions A2 , T1 , E02 , C12 , and C13 imply, by Lemmas 4.5.6 and 4.5.7, that (d) the cor."/ .0/ .ˇ/ ! jj .ˇ/ < 1 as " ! 0, and responding exponential moments converge, jj .0/ .ˇ/ 2 .1; 1/. Relations (d) and (e) imply that condition also give the relation (e) jj

."/ .t /. C6 holds for the distribution functions Gjj Note that conditions D22 and C6 imply condition D21 .

240

4 Perturbed semi-Markov processes ."/

The function 1Fj .t / is a forcing function for the renewal equation (4.2.1). Such functions are (f) non-decreasing, and (g) weakly converge by condition T1 . As was shown in the proof of Theorem 4.4.1, (f) and (g) imply that the functions 1  Fj."/ .t / converge locally uniformly in every point of the set Cj of positive continuity points of .0/ the limit function 1  Fj .t /. Note that, for the Lebesgue measure, m.C j / D 0. Thus ."/

condition F7 holds for the functions 1  Fj .t /. This shows that all conditions of Theorem 3.3.2 hold. By applying the asymptotic relation (2.2.18) in Theorem 3.3.2, we get the asymptotic relation (4.6.5) for i D j . We can apply Theorem 3.3.7 to prove the asymptotic relation (4.6.5) for the case where the initial state is i ¤ j . In this case, ."/ .t / is again a regenerative process with ."/ regeneration times j .n/ that are subsequent times of return of the process ."/ .t / into the state j . This process has a transition period, since the initial state differs from j but the shifted process ."/ .j."/ .1/ C t / is the standard regenerative process considered above. The renewal type relation (3.2.19) takes in this case the form of the renewal type relation (4.2.2). In this case again, A D fj g and the probability Pij."/ .t / plays the role of the probability PQ ."/ .t; A/. The distribution function 0 Gij."/ .t / replaces the function FQ ."/ .t / that generates this renewal type relation. ."/

The moment generating function 0 ij ./ coincides in this case with the moment generating function Q ."/ ./. Conditions A2 , T1 , E02 imply, by Lemmas 4.5.6 and 4.5.7, ."/ .0/ that (h) 0 ij ..0/ / ! 0 ij ..0/ / < 1 as " ! 0. Thus, condition C11 holds with .0/ the limit constant Q .0/ D 0  ..0/ /. ij

."/ The distribution function Gij .t / is regarded as the function FOQ ."/ .t /. Conditions 0 A2 , T1 , E2 , C12 , and C13 imply, by Lemma 4.5.7, that (i) condition C10 holds for these distribution functions, which implies that conditions C9 and D26 also hold. Finally, by applying the asymptotic relation (3.3.52) given in Theorem 3.3.7, we get the asymptotic relation (4.6.5) for the case where i ¤ j .

4.6.4 Quasi-stationary distributions for perturbed semi-Markov processes without non-recurrent-without-absorption states Let us assume that the following conditions hold: E03 : X1."/ D X0 D f1; : : : ; N g for all " small enough, say "  "06 ; ."/ .ı/ < 1, i ¤ 0, j 2 X, for all " small enough, say "  "07 ; C14 : pij

Note that the following condition implies condition C14 : C014 :

."/ ij .ı/

< 1, i ¤ 0, j 2 X, for all " small enough, say "  "07 ;

4.6 Mixed ergodic and large deviation theorems for semi-Markov processes

241

Let us also assume that the following conditions hold: C15 : There exists i ¤ 0 and ˇi 2 .0; ı, for ı defined in condition C14, such that: 0 'i."/ i .ˇi / 2 .1; 1/, for all " small enough, say "  "8 ; ."/ .t /, j ¤ 0, are non-arithmetic distribution functions for " small enough, N3 : 0 GN jj say "  "09 .

Set "010 D min."06 ; "07 ; "08 ; "09 /:

(4.6.6)

The following theorem is a direct corollary of Theorem 4.6.3. Theorem 4.6.4. Let conditions E03 , C14 , C15 , and N3 hold. Then, for every "  "010 , e

."/ t

."/

."/

Pi f."/ .t ."/ / D j; 0 > t g ! QQ ij .."/ / as t ! 1;

i; j ¤ 0:

(4.6.7)

Proof. Choose some "0  "010 and consider a model in which the transition proba."/ bilities Qij .t / of the semi-Markov processes ."/ .t /, t  0, do not depend on the 0

." / parameter " and coincide with the transition probabilities Qij .t /. In this case, conditions of Theorem 4.6.3 just reduce to conditions E03 , C14 , C15 , and N3 . The asymptotic relation (4.6.5) takes in this case the form of the asymptotic relation (4.6.7). In the same way one can prove relation (4.6.7) for any "  "010 .

Let ."/

j .."/ / D P

."/ QQ ij .."/ /

QQ ."/ .."/ / r ¤0  ir

;

."/

j 2 X1 :

(4.6.8)

Conditions A2 , T1 , E03 , C12 , and C13 , imply that QQ r."/ .."/ / 2 .0; 1/ for all " small enough, namely, "  min."06 ; "07 ; "08 /. This follows from Lemma 4.5.7. P ."/ ."/ Now, (a) j .."/ / > 0, j ¤ 0, and (b) j 2X ."/ j .."/ / D 1. Thus, formula 1 (4.6.8) defines a discrete probability distribution. This distribution is called a quasi-stationary distribution for the semi-Markov pro."/ cess ."/ .t /, t  0, with absorption times 0 . The following quasi-ergodic theorem is a direct corollary of Theorem 4.6.3. Theorem 4.6.5. Let conditions E03 , C14 , C15 , and N3 hold. Then, for "  "010 , ."/

Pi f."/ .t / D j=0 > t g D

e

."/ t

."/

Pi f."/ .t / D j; 0 > t g ."/

e ."/ t Pi f0 > t g ."/

! j .."/ / as t ! 1;

j ¤ 0:

(4.6.9)

242

4 Perturbed semi-Markov processes ."/

Relation (4.6.9) clarifies why it is natural to call the distribution j .."/ / a quasistationary distribution for the regenerative process ."/ .t / with absorption moments ."/ 0 . Formula (4.6.8) gives the quasi-stationary distribution in a form that seems to be dependent on the choice of the “initial” state i ¤ 0. The following lemma shows that ."/ the quasi-stationary probabilities j .."/ / defined in (4.6.8) are, actually, invariant with respect to the choice of the state i ¤ 0. Lemma 4.6.1. Let conditions E03 , C14 , C15 , and N3 hold. Then, for every "  "010 , ."/ j .."/ /

DP

."/ QQ ij .."/ /

QQ ."/ .."/ / r ¤0  ir

;

i; j ¤ 0:

(4.6.10)

Proof. Let us return to the proof of Theorem 4.6.3 for the case where the initial state i ¤ j , i; j 2 X1.0/ . The semi-Markov process ."/ .t /, more precisely, its modified version with the absorption state removed following the procedure described in Subsection 4.2.1, can be considered as a regenerative process. But the first regeneration time can be defined as the first time at which the semiMarkov process ."/ .t / hits the state j after first hitting some state k ¤ 0, and then the following regeneration times are defined to be the subsequent times of hitting the state j . More precisely, let us define the corresponding hitting times for the imbedded ."/ ."/ Markov chain n and the semi-Markov process ."/ .t /, respectively, by kj .1/ D ."/

min.n  k ."/

."/

."/

."/

."/

W n D j / and kj .1/ D ."/ .kj /, and then kj .n/ D min.n >

."/

."/

."/

kj .n  1/ W n D j / and kj .n/ D ."/ .kj .n// for n D 2; 3; : : : . The semi-Markov process ."/ .t / can be considered as a regenerative process with ."/ regeneration times kj .n/. This process has a transition period, since the initial state

differs from j but the shifted process ."/ .."/ .1/ C t / is the standard regenerative kj process considered above with regeneration times that are subsequent return times into the state j . Denote ."/ 0 Gikj .t /

."/

."/

."/

D Pi fkj .1/  t; kj .1/ < 0 g;

t  0;

i; j; k ¤ 0:

Let us apply Theorem 3.3.7. The renewal type relation (3.2.19) takes now the form of a renewal type relation different from (4.2.2). In this case again, A D fj g and the probability Pij."/ .t / is regarded as the probability PQ ."/ .t; A/. But the distribution function 0 G ."/ .t / replaces the distribution function FQ ."/ .t / that generates the following ikj

4.6 Mixed ergodic and large deviation theorems for semi-Markov processes

243

renewal type relation replacing (4.2.2): ."/ Pij .t /

where

D

."/ qikjj .t /

Z C

t 0

."/

."/

Pjj .t  s/0 Gikj .ds/;

t  0;

."/ qikjr .t / D Pi f."/ .t / D r; ."/ .1/ > t; ."/ 0 > t g; kj

(4.6.11)

t  0:

Using the regeneration property of the semi-Markov process ."/ .t / at hitting times, we can write the following formula for i; j; k ¤ 0: ."/ 0 Gikj .t /

."/ ."/ D 0 Gik .t /  0 Gkj .t /;

t  0:

(4.6.12)

."/

It follows from this formula that the distribution function 0 Gikj .t / has the following moment generating function for i; j; k ¤ 0: ."/ 0 ikj ./

."/

."/

D 0 ik ./  0 kj ./;

 2 R1 :

(4.6.13)

."/

The moment generating function 0 ikj ./ plays the role of the moment generating function Q ."/ ./. Conditions A2 , T1 , E0 imply, by Lemmas 4.5.6 and 4.5.7, that (a) ."/ .0/ 0 ikj . /

2

.0/

! 0 ikj ..0/ / < 1 as " ! 0. Thus, condition C11 holds with the limit .0/

.0/

.0/

constant Q .0/ D 0 ikj ..0/ / D 0 ik ..0/ /  0 kj ..0/ /. Denote ."/

."/

."/

Gikj .t / D Pi fkj .1/ ^ 0  t g;

t  0;

i; j; k ¤ 0:

Using the regeneration property of the semi-Markov process ."/ .t / at hitting times, we obtain the following formula for i; j; k ¤ 0: ."/

."/

."/

."/

."/

Gikj .t / D 0 Gikj .t / C k Gi0 .t / C 0 Gik .t /  j Gk0 .t /; t  0; i; j; k ¤ 0: (4.6.14) According to formulas (4.6.13) and (4.6.14), the moment generating functions ."/ Gikj .t / have the following moment generating function for i; j; k ¤ 0: ."/ ."/ ."/ ."/ ."/ ."/ ikj ./ D 0 ik ./  0 kj ./ C k i0 ./ C 0 ik ./  j k0 ./;

 2 R1 : (4.6.15)

."/

The distribution function Gikj .t / here is viewed as the distribution function

FOQ ."/ .t /. Conditions A2 , T1 , E02 , C12 , C13 imply, by Lemmas 4.5.6 and 4.5.7, that (b) these distribution functions satisfy condition C10 . This implies that conditions C9 and D26 hold.

244

4 Perturbed semi-Markov processes

Finally, by applying the asymptotic relation (3.3.52) given in Theorem 3.3.7 and assuming conditions E03 , C14 , C15 , and N2 to hold, we get the following asymptotic relation that is different from (4.6.5): ."/

Pi f."/ .t ."/ / D j; 0 > t ."/ g expf."/ t ."/ g .0/

.0/

.0/

! 0 ik ..0/ / 0 kj ..0/ /Qj ..0/ / as " ! 0;

i; k; j ¤ 0:

(4.6.16)

Comparing the asymptotic relations (4.6.5) and (4.6.16) we get an interesting relation, .0/ .0/ .0/ .0/ 0 ik . /0 kj . /

.0/

D 0 ij ..0/ /;

i; k; j ¤ 0:

(4.6.17)

."/

Let us consider a model in which the transition probabilities Qij .t / of the semiMarkov processes ."/ .t /, t  0, do not depend on the parameter ", more precisely, their values coincide for some particular value of the parameter "  "010 . In this case, conditions of Theorem 4.6.3 just reduce to conditions E03 , C14 , C15 , and N2 . Relation (4.6.17) takes in this case the following form for every "  "010 : ."/ ."/ ."/ ."/ 0 ik . /0 kj . /

D 0 ij."/ .."/ /;

i; k; j ¤ 0:

(4.6.18)

Relation (4.6.18) permits to complete the proof of Lemma 4.6.1. Using this formula we get for i; k; j ¤ 0 that ."/ j .."/ /

D P

QQ ij."/ .."/ /

QQ ."/ .."/ / r ¤0  ir

D P

DP

."/ ."/ Qj."/ .."/ / 0 ij . / ."/ ."/ Q r."/ .."/ / r¤0 0 ir . /

."/ ."/ ."/ ."/ ."/ Qj .."/ / 0 ik . / 0 kj . /

."/ ."/ ."/ ."/ ."/ Q r .."/ / r ¤0 0 ik . /0 kr . /

D P

."/ ."/ ."/ Qj .."/ / 0 kj . /

."/ ."/ ."/ Q r .."/ / r ¤0 0 kr . /

DP

."/ QQ kj .."/ /

QQ ."/ .."/ / r¤0  kr

:

(4.6.19)

The proof is complete. It is useful to note that formulas (4.6.18) and (4.6.19) hold without assuming that condition N3 holds. To see this, let us consider a semi-Markov process .0/ .t / with the transition prob.0/ .0/ .0/ abilities Qij .t / such that (a) X1 D X0 D f1; : : : ; N g; (b) ij .ı/ < 1, i ¤ 0, .0/

j 2 X; (c) there exist i ¤ 0 and ˇi 2 .0; ı for ı defined in (a) such that 'i i .ˇi / 2 .1; 1/. Condition N3 is not assumed to hold for the process .0/ .t /.

4.6 Mixed ergodic and large deviation theorems for semi-Markov processes

245

Now, let us choose and fix some state i 2 X0 and perturb the transition probabilities ."/ .0/ j 2 X, by replacing them with the convolutions Qij .t / D Q."/ .t /Qij .t /, ."/ j 2 X, where Q .t / D .t ="/ ^ 1, t  0, is the uniform distribution function on the interval Œ0; ". It is obvious that, for any " > 0, conditions (a)–(c) also hold for ."/ the semi-Markov processes ."/ .t /; t  0, with the transition probabilities Qij .t /. Obviously, condition N3 holds for the process ."/ .t /. Thus, by Lemma 4.6.1, the process ."/ .t / satisfies formulas (4.6.18) and (4.6.19) for every " > 0. Then, these formulas can be verified for the process .0/ .t / by letting " tend to 0 and using Lemma 4.5.6. Lemma 4.6.1 permits to vary the choice of the state i ¤ 0 in formula (4.6.10) as to simplify this formula. For example, one can choose the state i D j for every j ¤ 0 and get the following representation for quasi-stationary probabilities: .0/ .t /, Qij

."/

j."/ .."/ / D

Qj .."/ / ; P ."/ ."/ Qj .."/ / C r ¤0;j QQjr .."/ /

j ¤ 0:

(4.6.20)

."/ ."/ ."/ Here we used the identities 0 jj .."/ / D 1, j ¤ 0, and QQjj .."/ / D Qj .."/ /, j ¤ 0.

4.6.5 Convergence of quasi-stationary distributions for perturbed semiMarkov processes without non-recurrent-without-absorption states The following lemma gives asymptotic relations that permit to formulate conditions for convergence of quasi-stationary distributions of perturbed semi-Markov processes. ."/

Lemma 4.6.2. Let conditions A2 , T1 , E02 , C12 , and C13 hold. Then .˛/ Qj .."/ / ! .0/ ."/ .0/ Q ..0/ / as " ! 0, j ¤ 0; .ˇ/ QQ .."/ / ! QQ ..0/ / as " ! 0, i; j ¤ 0. j

ij

ij

Proof. It follows from Lemma 4.5.6 that, (a) under conditions of Lemma 4.6.2, the characteristic roots converge, ."/ ! .0/ as " ! 0, and that (b) 0  .0/ < ˇ  ı. Relations (a) and (b) imply that (c) there exist 0 < .0/ < ˇ 0 < ˇ and "011 D "011 .ˇ 0 / such that 0  ."/  ˇ 0 for "  "011 . Denote Z 1 ."/ N ."/ ./ D e s .1  Fj .s// ds;  2 R1 ; j ¤ 0: j 0

."/ .0/ Conditions T1 and C12 imply that (d) Nj ./ ! Nj ./ < 1 as " ! 0 for 0   < ı, j ¤ 0. Relation (d) implies that (e) there exists "012 > 0 such that N ."/ .ˇ 0 / < 1 for j ¤ 0 and "  min."0 ; "0 /. Consequently, relation (e) implies 11 12 j ."/ that (f) N ./ is a finite, non-negative, non-decreasing, and continuous function on j

246

4 Perturbed semi-Markov processes

the closed interval Œ0; ˇ 0  for j ¤ 0 and "  min."011 ; "012 /. It follows from (d) and (f) ."/ .0/ that (g) sup0ˇ 0 j Nj ./  Nj ./j ! 0 as " ! 0 for j ¤ 0. Relations (a), (c), (f), and (g) show that, for j ¤ 0, lim j Nj."/ .."/ /  Nj.0/ ..0/ /j   ."/ .0/ .0/ .0/ sup j Nj ./  Nj ./j C j Nj .."/ /  Nj ..0/ /j D 0: (4.6.21)  lim

"!0

"!0

0ˇ 0

Conditions T1 , C12 , and C13 also imply, by Lemma 4.5.6, that (h) 0 ij."/ ./ !

.0/ 0 ij ./

exists

< 1 as " ! 0 for 0   < ˇ, i; j ¤ 0. Relation (h) implies that (i) there

"013

."/

> 0 such that 0 ij .ˇ 0 / < 1 for i; j ¤ 0 and "  min."011 ; "012 ; "013 /. ."/

Hence, relation (i) implies that (j) the functions 0 ij ./ are finite, non-negative, nondecreasing, and continuous on the closed interval Œ0; ˇ 0  for i; j ¤ 0 and "  min."011 ; "012 ; "013 /. It follows from (h) and (j) that (k) sup0ˇ 0 j 0 ij."/ ./  .0/ 0 ij ./j

! 0 as " ! 0 for i; j ¤ 0. Relations (a), (c), (j), and (k) imply that, for i; j ¤ 0, ."/

."/

lim j0 ij .."/ /  0 ij ..0/ /j  sup j0 ij."/ ./  0 ij."/ ./j  lim

"!0

"!0

0ˇ 0

 C j0 ij.0/ .."/ /  0 ij.0/ ..0/ /j D 0:

(4.6.22)

."/

Formula (4.6.3), which defines the coefficients Qj .."/ /, and relations (4.6.21) and (4.6.22) imply that, for j ¤ 0, Qj."/ .."/ / D

N ."/ .."/ /

j ."/ ."/ 0 jj . /

! Qj.0/ ..0/ / D

N .0/ ..0/ /

j .0/ .0/ 0 jj . /

as " ! 0;

(4.6.23)

and, for i; j ¤ 0, QQ ij."/ .."/ / D 0 ij."/ .."/ / .0/

N ."/ .."/ /

j ."/ ."/ 0 jj . / .0/

! QQj ..0/ / D 0 ij ..0/ / The proof is complete.

N .0/ ..0/ /

j .0/ .0/ 0 jj . /

as " ! 0:

(4.6.24)

4.6 Mixed ergodic and large deviation theorems for semi-Markov processes

247

The following theorem is a direct corollary of formula (4.6.8) and Lemma 4.6.2. It gives an asymptotic relation that supplements the asymptotic relations (4.6.5) and (4.6.7). Theorem 4.6.6. Let conditions A2 , T1 , E02 , C12 , and C13 hold. Then ."/

.0/

j .."/ / ! j ..0/ / as " ! 0; j ¤ 0:

(4.6.25)

4.6.6 Quasi-stationary exponential asymptotics for perturbed semi-Markov processes with non-recurrent-without-absorption states Let us now consider a model for which condition E002 holds. Define 8 < Q r."/ .."/ / for r 2 X ."/ , 1 ."/ ."/ Q r . / D ."/ : 0 for r 2 X2 , and ."/ QQ ir .."/ / D

(4.6.26)

8 < 0  ."/ .."/ /Q r."/ .."/ / for i ¤ 0, r 2 X ."/ , 1 ir : 0 ."/

."/

for i ¤ 0, r 2 X2 ,

(4.6.27)

."/

where the coefficients Q r .."/ /, r 2 X1 are defined by formula (4.6.3). Theorem 4.6.7. Let conditions A2 , T1 , E002 , C12 , C13 , and N1 hold. Then, for any 0  t ."/ ! 1 as " ! 0, ."/ g Pi f."/ .t ."/ / D j; ."/ .0/ 0 >t ! QQ ij ..0/ / as " ! 0; expf."/ t ."/ g

i; j ¤ 0:

(4.6.28) .0/

Proof. Let us first consider that case where the initial state is i D r and r D j 2 X1 . The first part of the proof, where conditions D22 , C6 , and F7 are verified, repeats the same part of the proof of Theorem 4.6.5. The only difference is that the reference to condition E02 should be replaced with an equivalent reference to condition E002 . Thus, all conditions of Theorem 3.3.2 hold. By applying the asymptotic relation (2.2.18) given in Theorem 3.3.2, we get the asymptotic relation (4.6.28) for the case .0/ where i D r and r D j 2 X1 . Let us now consider the case where the initial state i ¤ r; 0 and r D j 2 X1.0/ . As above, the renewal type relation (3.2.19) takes in this case the form of the re."/ newal equation (4.2.2) with the distribution function 0 Gij .t /, being regarded as the distribution function FQ ."/ .t / that generates this renewal type relation, and the mo."/ ment generating function 0 ij ./ replacing the moment generating function, Q ."/ ./.

248

4 Perturbed semi-Markov processes ."/

Conditions A2 , T1 , E002 imply, by Lemmas 4.5.6 and 4.5.7, that (a) 0 ij ..0/ / !

.0/ .0/ 0 ij . / < 1 as .0/ Q .0/ D 0 ij ..0/ /.

" ! 0. Thus, condition C11 holds with the limit constant

."/ The distribution function Gij .t / plays the role of the distribution function FOQ ."/ .t /. Conditions A2 , T1 , E02 and C12 and C13 imply, by Lemmas 4.5.7, that (b) condition C10 holds for these distribution functions. This condition implies that conditions C9 and D26 are satisfied. Finally, by applying the asymptotic relation (3.3.52) given in Theorem 3.3.7, we .0/ get the asymptotic relation (4.6.28) for the case where i ¤ r; 0 and r D j 2 X1 . .0/ It still remains to prove validity of the asymptotic relation (4.6.28) for r 2 X2 . In this case, we can choose some state j 2 X1.0/ and use the renewal relations (4.2.3) and (4.2.6), instead of the renewal relations (4.2.1) and (4.2.2). .0/ To this end, let us first consider the case where the initial state is i D j 2 X1 , .0/ r 2 X2 . The renewal equations (4.2.3) and (4.2.1) have the same distribution function ."/ ."/ 0 Gjj .t / that generates these equations, and the distribution function Gjj .t / takes place of the distribution function FO ."/ .t /. This shows that conditions D22 and C6 hold. However, the renewal equation (4.2.3) has a forcing function that differs from the forcing function for renewal equation (4.2.1). Condition F7 was verified, in this case, in the proof of Theorem 4.4.2. In particular, it was shown that the limit forcing function .0/ .0/ is qjr .t / D 0, t  0. Thus, for the set A D frg, r 2 X2 , we have

R1

Q

.0/

.A/ D

.0/ s q .0/ .s/ ds jr 0 e R1 .0/  s G .0/ .ds/ 0 jj 0 se

D Q r.0/ ..0/ / D 0:

(4.6.29)

Finally, by applying the asymptotic relation (3.3.52) given in Theorem 3.3.7, we .0/ .0/ get the asymptotic relation (4.6.28) for the case where i D j 2 X1 , r 2 X2 . The last case to be considered is where the initial state i ¤ j; 0, j 2 X1.0/ , and r 2 X2.0/ . The renewal relation (3.2.19) takes now the form of the renewal relation (4.2.6) that should be used instead of the renewal relation (4.2.2). The renewal relations (4.2.6) ."/ and (4.2.2) have the same distribution function 0 Gij .t / that generates these equations ."/

and the same distribution function Gij .t / which replaces the distribution function FOQ ."/ .t /. ."/

The moment generating function 0 ij ./ coincides in this case with the moment generating function Q ."/ ./. Conditions A2 , T1 , E002 imply, by Lemmas 4.5.6 and 4.5.7, ."/ .0/ that (c) 0 ij ..0/ / ! 0 ij ..0/ / < 1 as " ! 0. Thus, condition C11 holds with the .0/ limit constant Q .0/ D 0  ..0/ /. ij

4.6 Mixed ergodic and large deviation theorems for semi-Markov processes

249

."/ The distribution function Gij .t / is regarded as the distribution function FOQ ."/ .t /. Conditions A2 , T1 , E002 , and C12 and C13 imply, by Lemmas 4.5.7, that (d) condition ."/ C10 holds for the distribution function Gij .t / that plays the role of the distribution function FOQ ."/ .t /. This condition implies that conditions C and D are verified. 9

26

Finally, by applying the asymptotic relation (3.3.52) given in Theorem 3.3.7, we get .0/ .0/ the asymptotic relation (4.6.28) for the case where i ¤ j; 0, j 2 X1 , and r 2 X2 .

4.6.7 Quasi-stationary distributions for perturbed semi-Markov processes with non-recurrent-without-absorption states Let us consider the general case where the following condition (introduced in Subsection 4.4.5) holds: ."/

E3 : X1 ¤ ¿ for the semi-Markov process ."/ .t / for " small enough, say "  "6 . We also assume to hold the following condition that should replace condition C15: C16 : There exists i 2 X1."/ and ˇi 2 .0; ı for ı defined in condition C14 such that ."/

(a): 0 i i .ˇi / 2 .1; 1/ for all " small enough, say "  "014 ; ."/

.0/

(b): 0 ki .ˇi / < 1, k 2 X2

for all "  "014 .

Let "015 D min."6 ; "07 ; "09 ; "014 /:

(4.6.30)

The following theorem is an analogue of Theorem 4.6.5. Theorem 4.6.8. Let conditions E3 , C14 , C15 , and N2 hold. Then, for every "  "015 , e

."/ t

QQ ."/ .."/ / as t ! 1; Pi f."/ .t ."/ / D j; ."/ 0 > tg !  ij

i; j ¤ 0:

(4.6.31)

Proof. Choose some "0  "015 and consider a model in which the transition probabili."/ ties Qij .t / of semi-Markov processes ."/ .t /, t  0, do not depend on the parameter ."0 /

" and coincide with the transition probabilities Qij .t /. In this case, conditions of Theorem 4.6.7 just reduce to conditions E3 , C14 , C15 , and N2 . The asymptotic relation (4.6.28) take the form of the asymptotic relation (4.6.31). Relation (4.6.31) can be proved in the same way for any "  "015 . Denote ."/

j .."/ / D P

."/ QQ ij .."/ /

QQ ."/ .."/ / r ¤0  ir

."/

where QQ ij .."/ /, i; j ¤ 0, are defined by (4.6.27).

;

j ¤ 0;

(4.6.32)

250

4 Perturbed semi-Markov processes

."/ Conditions A2 , T1 , and E3 , C12 , and C13 , imply that QQ r .."/ / 2 .0; 1/ for all " small enough, namely, "  min."6 ; "07 ; "014 /. This follows from Lemma 4.5.7. ."/ ."/ ."/ ."/ In this case, (a) j .."/ / > 0, j 2 X1 ; (b) j .."/ / D 0, j 2 X2 ; (c) P ."/ ."/ ."/  j . / D 1. Thus, formula (4.6.32) defines a discrete probability distrij 2X1 bution. This distribution is called a quasi-stationary distribution for the semi-Markov pro."/ cess ."/ .t /, t  0, with absorption times 0 . The following quasi-stationary theorem is a direct corollary of Theorem 4.6.8.

Theorem 4.6.9. Let conditions E3 , C14 , C15 , and N2 hold. Then, for "  "015 , ."/

Pi f."/ .t / D j=0 > t g D

e

."/ t

."/

Pi f."/ .t / D j; 0 > t g ."/

e ."/ t Pi f0 > t g ."/

! j .."/ / as t ! 1;

j ¤ 0:

(4.6.33)

Relation (4.6.33) clarifies why it is natural to call the distribution j."/ .."/ / a quasistationary distribution for the semi-Markov process ."/ .t / with the absorption mo."/ ments 0 . Formula (4.6.32) defines the quasi-stationary distribution in the form that seems to be dependent on the choice of the ”initial” state i ¤ 0. The following lemma permits to prove that the quasi-stationary probabilities j."/ .."/ / defined in (4.6.32) are, actually, independent of a choice of the state i ¤ 0. Lemma 4.6.3. Let conditions E3 , C14 , C15 , and N2 hold. Then, for every "  "015 , ."/ j .."/ /

DP

QQ ij."/ .."/ / QQ ."/ .."/ / r ¤0  ir

;

i; j ¤ 0:

(4.6.34)

.0/

Proof. The case j 2 X2 is trivial. Indeed, formula (4.6.26) implies in this case that ."/ .0/ QQ ij .."/ / D 0 for such j . In the case j 2 X1 , the proof repeats the proof of Lemma .0/

4.6.1. The only difference is that now one should take the states i ¤ j; 0, j; k 2 X1 , and then repeat the proof of Lemma 4.6.1. In particular, one can refer to conditions A2 , T1 , E002 , instead of conditions A2 , T1 , E02 , when applying Lemmas 4.5.6 and 4.5.7 to get, with the use conditions A2 , T1 , E002 , and N2 , the following formula for every .0/ i ¤ j; 0, j; k 2 X1 : ."/ 0 ikj ./

."/

."/

D 0 ik ./  0 kj ./;

 2 R1 :

(4.6.35) ."/

Then one considers a model in which the transition probabilities Qij .t / of the semi-Markov processes ."/ .t /; t  0, do not depend on the parameter ", more precisely, their values coincide for some particular value of the parameter "  "015 . In

4.6 Mixed ergodic and large deviation theorems for semi-Markov processes

251

this case, condition E3 coincides with either condition E02 or E002 . Formula (4.6.17) in the first case and formula (4.6.35) in the second case yield, for every i ¤ j; 0; j; k 2 X1.0/ , ."/ ."/ ."/ ."/ 0 ik . /0 kj . /

D 0 ij."/ .."/ /:

(4.6.36)

Using relation (4.6.36) we prove formula (4.6.34) repeating transformations given in relation (4.6.19).

4.6.8 Convergence of quasi-stationary distributions of perturbed semiMarkov processes with non-recurrent-without-absorption states The following lemma is an analogue of Lemma 4.6.2 for the model of perturbed semiMarkov processes with non-recurrent-without-absorption states. ."/

Lemma 4.6.4. Let conditions A2 , T1 , E002 , C12 , and C13 hold. Then .˛/ Qj .."/ / ! .0/ ."/ .0/ Q ..0/ / as " ! 0; j ¤ 0; .ˇ/ QQ .."/ / ! QQ ..0/ / as " ! 0, i; j ¤ 0. j

ij

ij

.0/

."/

Proof. Note that, under conditions A2 , T1 , and E002 , we have X1 X1 for all " small enough. Taking into account this fact and repeating the first part of the proof of Lemma 4.6.2 we see that, for j 2 X1.0/ , ."/

.0/

Qj .."/ / ! Qj ..0/ / as " ! 0:

(4.6.37)

Then, using relation (4.6.37) we get X ."/ X ."/ Qj .."/ / D 1  Qj .."/ / j 2X2.0/

j 2X1.0/

!1

X

Qj.0/ ..0/ / D 0 as " ! 0:

(4.6.38)

.0/

j 2X1

The proof of the following relation repeats the corresponding part in the proof of Lemma 4.6.2: ."/ ."/ 0 ij . /

.0/

! 0 ij ..0/ / as " ! 0;

i ¤ 0;

.0/

j 2 X1 :

(4.6.39)

Relations (4.6.37), (4.6.38), and (4.6.39) prove the lemma. The following theorem is a direct corollary of Lemma 4.6.4. Theorem 4.6.10. Let conditions A2 , T1 , E002 , C12 , and C13 hold. Then ."/

.0/

j .."/ / ! j ..0/ / as " ! 0;

j ¤ 0:

(4.6.40)

252

4 Perturbed semi-Markov processes

4.6.9 Large deviation asymptotics for distributions of absorption times Denote

QQ i."/ .."/ / D

X

."/ ."/ . /; QQ ir

i ¤ 0:

."/ r 2X1

The following theorem is an obvious corollary of Theorems 4.6.3 and 4.6.7. Theorem 4.6.11. Let conditions A2 , T1 , E2 , C12 , C13, and N1 hold. Then, for any 0  t ."/ ! 1 as " ! 0, ."/

Pi f0 > t ."/ g ! QQ i.0/ ..0/ / as " ! 0; expf."/ t ."/ g

i ¤ 0:

(4.6.41)

Proof. Relation (4.6.41) is a direct corollary of asymptotic relations (4.6.5) and (4.6.28) given, respectively, in Theorems 4.6.3 and 4.6.7. It can be obtained by summing over r ¤ 0 the pre-limit and limit expressions in the left and the right-hand sides of (4.6.5) or (4.6.28) depending on whether condition E02 or E002 holds. Let us comment on the asymptotic relation (4.6.41) given in Theorem 4.6.11. We consider the main case where condition E2 is realised in the form of condition E02 . Asymptotic relation (4.6.41) becomes trivial if condition A4 holds, i.e., if P ."/ ."/ D 1g D 1, i¤0 pi0 D 0 for all " small enough. Indeed, in this case, Pi f i ¤ 0, for all such " and both expressions in the left and the right-hand sides of (4.6.41) are equal to 1. Let us assume, additionally to conditions of Theorem 4.6.11, the following condition that guarantees that Pi f."/ < 1g D 1, i ¤ 0 for every " > 0: A5 :

P

."/

i2X1

."/ pi0 > 0 for " > 0.

Condition A5 is equivalent to the assumption that (a) ."/ > 0 for " > 0. Also, as follows from Lemma 4.5.6, conditions A2 , T1 , E02 , C12 , and C13 imply that (b) ."/ ! .0/ as " ! 0. Note also that, according to Lemma 4.2.2, conditions T1 and E02 imply that X1."/ D X0 D f1; : : : ; N g for all " small enough, say "  "016 . There are two alternative cases. One is where (i) condition A3 holds or, equivalently, .0/ D 0 and another one is where (ii) condition A3 does not hold or, equivalently, .0/ > 0. Let us first consider case (i). By Lemma 4.5.8, condition C13 can be omitted. Relation (b) in this case takes the form (b0 ) ."/ ! .0/ D 0 as " ! 0. .0/ .0/ Also, in this case, (c) QQ i .0/ D 1, i ¤ 0. Indeed, (d) 0 ij .0/ D 1, i; j ¤ 0. .0/

.0/

.0/

Also, (e) Qj .0/ D j .0/, j ¤ 0, where j ; j ¤ 0, are stationary probabilities

4.6 Mixed ergodic and large deviation theorems for semi-Markov processes

given by formula (4.4.7). Thus, using (d) and (e) we get X X .0/ Q r.0/ .0/ D r.0/ .0/ D 1: QQ i.0/ .0/ D 0 ir .0/  .0/ r 2X1

253

(4.6.42)

.0/ r2X1

Let us first choose t ."/ ! 1 such that ."/ t ."/ ! 0 as " ! 0. Then the asymptotic relation (4.6.41) takes the following form for i ¤ 0: Pi f."/ > t ."/ g ! 1 as " ! 0:

(4.6.43)

Relation (4.6.43) implies that, in case (f), the stopping moments ."/ are asymptotically stochastically unbounded random variables as " ! 0, i.e., for i ¤ 0, lim Pi f."/ > T g D 1;

"!0

(4.6.44)

T > 0:

Let us take t ."/ D t =."/ ! 1 as " ! 0, where t > 0. Asymptotic relation (4.6.41) takes the following form for i ¤ 0: Pi f."/ ."/ > t g ! 1 as " ! 0; expft g

(4.6.45)

t > 0:

This is a weak convergence asymptotic relation. Indeed, relation (4.6.45) means d

that, for any initial state i ¤ 0, the random variables converge, ."/ ."/ ! 0 as " ! 0, where 0 is an exponentially distributed random variable with parameter 1. Note that (4.6.45) is consistent with the asymptotic relation (4.4.40) given in Theorem 4.4.6. Indeed, as was mentioned in Remark 4.6.1, by Lemma 1.4.2 with conditions ."/ ."/ A2 , T1 , E2 C12 , A3 fulfilled, (g) the characteristic root satisfies ."/  jfj 0 = 0 Mjj as " ! 0. According to Lemmas 4.2.4 and 4.3.4, and formula (4.4.11), we have in ."/ ."/ =m.0/ as " ! 0, where f ."/ and m.0/ are this case that (h) jfj."/ 0 = 0 Mjj  f defined in formulas (4.3.5) and (4.4.29). If t ."/ D t =."/ , then (g) and (h) imply that ."/ t ."/ ! =m.0/ as " ! 0. In these cases, asymptotic relation (4.6.41) given in The."/ orem 4.6.11 reduces to the asymptotic relation (i) Pi ff ."/ 0 > t g ! expft =m.0/ g as " ! 0; t > 0, i ¤ 0. This relation is equivalent to the asymptotic relation (4.4.40) given in Theorem 4.4.6. It should be noted that the asymptotic relation (4.4.40) was obtained under conditions weaker than the asymptotic relation (4.6.41). We can, however, take t ."/ D tQ."/ =."/ where tQ."/ ! 1 as " ! 0. In this case, (4.6.41) takes the form of a large deviation asymptotic relation for i ¤ 0, Pi f."/ ."/ > tQ."/ g ! 1 as " ! 0: expftQ."/ g

(4.6.46)

Asymptotic relation (4.6.46) is much stronger than asymptotic relation (i). This relation implies that the asymptotic relative error, if the tail probabilities

254

4 Perturbed semi-Markov processes

Pi f."/ ."/ > tQ."/ g are approximated with the corresponding tail probabilities for limit exponential distribution, equals 0 for any tQ."/ ! 1. This is achieved by choosing very well-fitted normalising coefficients ."/ . A payment for this is the need to use the Cram´er type condition C12 . A drawback of (4.6.46) is that the normalising coefficient ."/ is given as a root of the nonlinear equation (4.5.1). The question arises if it is possible to replace ."/ in (4.6.46), for example, by a simpler coefficient f ."/ =m.0/ and to rewrite this relation in the form (j) Pi ff ."/ ."/ > tQ."/ m.0/ g= expftQ."/ m.0/ g ! 1 as " ! 0; t > 0, i ¤ 0? The answer is partly in the affirmative. Relation ."/  f ."/ =m.0/ means that ."/ D f ."/ =m.0/ C o.f ."/ /. Thus, (4.6.46) implies that (j) Pi ff ."/ ."/ > tQ."/ m.0/ g  expf."/ tQ."/ m.0/ =f ."/ g  expf.f ."/ =m.0/ C o.f ."/ //tQ."/ m.0/ =f ."/ g. Therefore, the suggested replacement would additionally require to assume that (k) tQ."/ o.f ."/ /=f ."/ ! 0, i.e., to restrict the rate of tQ."/ approaching 1. The questions related to the use of normalising coefficients that are given in a more explicit form will be investigated in the next chapter where we shall get asymptotic expansions for the roots ."/ in asymptotic power series. Let us now consider the case (ii). Relation (b) takes in this case the form (b00 ) ."/ ! .0/ > 0 as " ! 0. Also, (l) it is possible that QQ i.0/ ..0/ / ¤ 1, at least for some i ¤ 0. Indeed, it can .0/ .0/ .0/ happen that (m) 0 ij ..0/ / ¤ 1 at least for some i; j ¤ 0, and (e) Qj ..0/ / ¤ j .0/ at least for some j ¤ 0. Moreover, the coefficients QQ i ..0/ / can depend on i , i.e., the limit expression in the asymptotic relation (4.6.41) can depend on the choice of the initial state i ¤ 0 used in the left-hand side of this relation. However, we have the .0/ .0/ .0/ following formula: (m) QQ i ..0/ /=QQj ..0/ / D 0 ij ..0/ /; i; j ¤ 0. Indeed, using formula (4.6.18) we get for i; j ¤ 0, that P .0/ .0/ .0/ .0/ .0/ .0/ Q r ..0/ / QQ i ..0/ / r ¤0 0 ij . /0 jr . / D D 0 ij.0/ ..0/ /: (4.6.47) P .0/ .0/ ."/ ."/ ."/ ."/ Q Q r . / Qj . / r ¤0 0 jr . / Let us introduce a distribution function for absorption times by ."/

."/

Gi .t / D Pi f0  t g; and the corresponding Laplace transforms Z 1 ."/ ."/ N e s Gi .ds/; i ./ D 0

t  0;

i ¤ 0;

  0;

i ¤ 0:

By using the regeneration property possessed by the semi-Markov process ."/ .t /, t  0, at the time of first hitting into any state, we can write the following renewal type relation for i ¤ 0: ."/ ."/ Gi."/ .t / D i Gi0 .t / C 0 Gi."/ i .t /  Gi .t /;

t  0:

(4.6.48)

4.6 Mixed ergodic and large deviation theorems for semi-Markov processes

255

In terms of the Laplace transform, this formula takes the following form for i ¤ 0: ."/ N ."/ N i."/ ./ D i Ni0 ./ C 0 Ni."/ i ./i ./;

  0:

(4.6.49)

Note that if condition E02 is verified, the assumption .0/ > 0 is equivalent to the .0/ ."/ assumption that (n) 0 fi i < 1, i ¤ 0. Under condition T1 , we have (0) 0 Ni i .0/ ! .0/ N ."/ .0/ D 0 f ."/ < 1, i ¤ 0, for " small 0 fi i as " ! 0, i ¤ 0 and, thus, (p) 0  ii ii enough, say "  "017 . Taking into account relation (p) we can re-write relation (4.6.49) in the following form for i ¤ 0 and "  min."016 ; "017 /: N i."/ ./ D

1

N ."/ i i0 ./ ; ."/  0 N i i ./

  0:

(4.6.50)

."/ .0/ Using Lemmas 4.2.1 and 4.2.3, we have (q) i N i0 ./ ! i N i0 ./ as " ! 0 for ."/ .0/   0 and i ¤ 0, and (r) 0 Ni i ./ ! 0 N i i ./ as " ! 0 for   0 and i ¤ 0. ."/ .0/ Relations (q) and (r) imply that (s) N i ./ ! N i ./ as " ! 0 for   0 and i ¤ 0. Thus, for i ¤ 0,

Gi."/ ./ ) Gi."/ ./ as " ! 0:

(4.6.51) ."/

Relation (4.6.51) implies that, in the case (ii), the stopping moments 0 are asymptotically stochastically bounded random variables as " ! 0, i.e., lim lim Pi f."/ 0 > T g D 0;

T !1 "!0

i ¤ 0:

(4.6.52)

Under conditions of Theorem 4.6.11, relation (4.6.41) yields, in case (ii), that for any 0  t ."/ ! 1 as " ! 0, ."/

Pi f0 > t ."/ g ! QQ i.0/ ..0/ / as " ! 0; expf."/ t ."/ g

i ¤ 0:

(4.6.53)

Note that in case (ii), we also have that ."/ ! .0/ > 0, thus the tail probabilities for stopping moments have exponential decay with parameter ."/ that is separated from zero asymptotically as " ! 0. In conclusion, let us show what form will the corresponding conditions take in the ."/ .0/ case where ."/ .t /  .0/ .t /, t  0, and, therefore, 0  0 for all "  0, i.e., the processes and the absorption moments do not depend on the parameter ". Conditions of Theorem 4.6.11 reduce to the following assumptions: (t) condi.0/ tion E02 holds, i.e., the class of recurrent-without-absorption states is X1 D X0 D .0/ .0/ f1; : : : ; N g; (u) ij .ı/ < 1, i ¤ 0, j 2 X; (v) 0 i i .ˇi / 2 .1; 1/ for some

256

4 Perturbed semi-Markov processes

P .0/ .0/ 0 < ˇi  ı and i ¤ 0; (w) i¤0 pi0 > 0, i.e., .0/ > 0; (x) 0 Gi i .t /, i ¤ 0, are non-arithmetic distribution functions, i.e., condition N1 holds. Relation (4.6.41) takes the following form: .0/

Pi f0 > t g QQ i.0/ ..0/ / expf.0/ t g

! 1 as t ! 1;

i ¤ 0:

(4.6.54)

Asymptotic relation (4.6.54) means that the tail probabilities Pi f.0/ > t g have 0 exponential decay and can be approximated with the corresponding tail probabilities .0/ .0/ for an exponential type distribution with the tail probabilities QQ i ..0/ /e  t . The asymptotic relative error of such an approximation equals 0 as t ! 1. The asymptotic relation (4.6.41) given in Theorem 4.6.11 can be commented on in a similar way when condition E2 is realised in the form of condition E002 , i.e., when the limit semi-Markov process has a non-empty class of non-recurrent-without-absorption states. The remarks above allow us to interpret the mixed asymptotic relations given in Theorems 4.6.1– 4.6.3 and 4.6.7 as mixed ergodic and large deviation theorems for perturbed semi-Markov processes with absorption. These theorems, together with Theorems 4.6.4 – 4.6.6, 4.6.8– 4.6.11, describe the so-called pseudo-stationary and quasi-stationary phenomena for perturbed semi-Markov processes. We would like to mention once more that there is an essential difference between the asymptotic relations given in Theorems 4.6.1– 4.6.11 for the cases where (i) .0/ D 0, or (ii) where .0/ > 0. In the first case, absorption is impossible for the limit semi-Markov process .0/ .t /, ."/ which implies asymptotic stochastic unboundedness of absorption times 0 as " ! 0. The corresponding asymptotic relations describe the pseudo-stationary phenomenon for perturbed semi-Markov processes. In the second case, absorption is possible for the limit semi-Markov process .0/ .t /, ."/ which implies asymptotic stochastic boundedness of the absorption times 0 as " ! 0. The corresponding asymptotic relations describe the quasi-stationary phenomenon for perturbed semi-Markov processes.

Chapter 5

Nonlinearly perturbed semi-Markov processes

Chapter 5 contains asymptotic results which improve the mixed ergodic theorems (for semi-Markov processes) and large deviation theorems (for absorption times), obtained in Chapter 4, to the case of nonlinearly perturbed semi-Markov processes with a finite set of states. As in the preceding chapter, we consider semi-Markov processes with a finite set of states, X D f0; 1; : : : ; N g, and the absorption state 0. The first time of hitting this state is the absorption time for the corresponding semi-Markov process. The above asymptotic results are given in the form of exponential expansions for joint distributions of positions of semi-Markov processes and absorption times. These expansions reduce, in the simplest zero-order case, to the mixed ergodic and large deviation theorems given in Chapter 4. They are obtained by applying the corresponding results, discussed in Chapter 3, on the exponential expansions for regenerative processes and mixed ergodic and large deviation theorems obtained in Chapter 4 for nonlinearly perturbed semi-Markov processes. It also contains results related to asymptotic expansions for quasi-stationary distributions and asymptotic expansions and their pivotal properties for potential matrices. We consider the same model of perturbed semi-Markov processes as in Chapter 4. However, in Chapter 5, we impose stronger perturbation conditions on the transi."/ tion probabilities of the imbedded Markov chains pij and the semi-Markov processes ."/

."/

Qij .t /. We assume that these transition probabilities pij and the power or the mixed ."/

power-exponential moments of transition probabilities Qij .t / (which are improper distribution functions in argument t ) can be expanded in a power series with respect to " up to and including the order k. As above, the relationship between the rate with which " tends to zero and the time t tends to infinity has a delicate influence upon the results. Without loss of generality we assume that the time t D t ."/ is a function of the parameter ". The balance between the rate of perturbation and the rate of growth of time is characterised by the following condition that is assumed to hold for some 1  r  k, "r t ."/ ! r < 1 as " ! 0:

(5.0.1)

With the above assumptions and under some natural additional conditions of ."/ Cram´er type for the transition probabilities Qij .u/, we obtain the asymptotic

258

5

Nonlinearly perturbed semi-Markov processes

relation ."/

Pi f."/ .t ."/ / D j; 0 > t ."/ g .0/ ! QQ ij ..0/ /e r ar as " ! 0: (5.0.2) .0/ r 1 ."/ expf. C a1 " C    C ar 1 " /t g We derive an equation for finding the coefficient .0/ , systems of linear equations .0/ to determine QQ ij ..0/ /, i; j ¤ 0, and finally an explicit recurrence algorithm for calculating the coefficients and a1 ; : : : ; ak as rational functions of the coefficients in the expansions for the moments of the transition probabilities. The results known in the literature and related to the asymptotics given in (5.0.2) mainly cover the case k D 1, which corresponds to a model of linearly perturbed processes. Relation (5.0.2) for k > 1 corresponds to a model of nonlinearly perturbed processes. This is one of the main new elements in our results. One more new element is a unified treatment of different types of asymptotics. Both the pseudo- and the quasi-stationary cases are covered by the same analytical approach. Relation (5.0.2) can be interpreted as a new kind of ergodic and mixed large deviation theorems for nonlinearly perturbed semi-Markov processes. We obtain mixed large deviation and ergodic theorems for nonlinearly perturbed semi-Markov processes using exponential expansions in mixed ergodic and large deviation theorems developed in Chapter 3 for nonlinearly perturbed regenerative processes. We use the fact that any semi-Markov processes is a regenerative process with the times of returning to some fixed state being regeneration times. The time of absorption (the first time of hitting the absorbing state) is a stopping time. A semi-Markov process is characterized by prescribing its transition probabilities. It is natural to formulate the perturbation conditions in terms of these transition probabilities. However, the conditions and expansions formulated for regenerative processes are specified in terms of expansions for the moments of regeneration times. Therefore, the corresponding asymptotic expansions for absorption probabilities and the moments of return and hitting times for perturbed semi-Markov processes must be developed. We construct asymptotic expansions for hitting probabilities, and for power moments and mixed power-exponential moments of hitting times using a procedure that is based on recursive systems of linear equations for hitting probabilities and moments of hitting times as well as provide these expansions with a detailed analysis of their pivotal properties. These results seem to be important by themselves. We also obtain nonlinear asymptotic expansions for quasi-stationary distributions of nonlinearly perturbed semi-Markov processes, j."/ .."/ / D j.0/ ..0/ / C gj Œ1" C    C gj Œk"k C o."k /:

(5.0.3)

We give an explicit recurrence algorithm for calculating the coefficients and gj Œ1; : : : ; gj Œk as rational functions of the coefficients in the expansions for the moment functionals of transition probabilities.

5.1 Asymptotic expansions for absorption and hitting probabilities

259

In the case of the asymptotic relation (5.0.3), the results known in the literature are related to the model of linearly perturbed Markov chains and the first order asymptotics. New elements in our studies are nonlinear perturbations and nonlinear higher order asymptotic expansions of type (5.0.3) for quasi-stationary and usual stationary distributions of the perturbed semi-Markov processes. As a particular but important example, we consider the model of continuous time Markov chains with absorption. In this case, it is more natural to formulate the perturbation conditions in terms of generators of the Markov chains. Here, an additional step in the algorithm is needed, since the initial perturbation conditions for generators must be expressed in terms of the moments for the corresponding semi-Markov transition probabilities before the basic algorithm can be applied. The first three sections are devoted to studies of asymptotic expansions for hitting probabilities, and for power moments and mixed power-exponential moments of hitting times for perturbed semi-Markov processes. Lemmas 5.1.1, 5.1.2, 5.1.3, and 5.1.4 present asymptotic expansions and describe pivotal properties of solutions of nonlinearly perturbed “hitting type” systems of linear equations. Lemma 5.2.1 gives a recursion algorithm for constructing asymptotic expansions for power moments of hitting times while Lemmas 5.3.5 and 5.3.6 present similar results for mixed powerexponential moments of hitting times. Theorems 5.4.1, 5.4.2, 5.4.3, and 5.4.4 give exponential expansions in mixed ergodic and large deviation theorem for perturbed semi-Markov processes. The first two theorems cover the case of the pseudo-stationary asymptotics, where the absorption probabilities are asymptotically small. The latter two theorems deal with the case of the quasi-stationary asymptotics for the absorption probabilities that are asymptotically separated from zero. Theorem 5.5.1 gives higher order asymptotic expansions for stationary distributions of ergodic nonlinearly perturbed semi-Markov processes. Theorems 5.5.4 and 5.5.5 give higher order asymptotic expansions for quasi-stationary distributions in the case of nonlinearly perturbed semi-Markov processes. The mixed ergodic and limit/large deviation theorems obtained in Chapters 4 and 5 as well as the exponential expansions provided by these theorems are used in Chapter 6 as a tool for an analysis of quasi-stationary phenomena in models of population dynamics, epidemic models, and highly reliable queueing systems.

5.1 Asymptotic expansions for absorption and hitting probabilities In this section, we describe a recursion algorithm for the asymptotic analysis of hitting and absorption probabilities for nonlinearly perturbed semi-Markov processes. The

260

5

Nonlinearly perturbed semi-Markov processes

algorithm is important in its own right and makes an essential part of the proof of our main results concerning the semi-Markov processes.

5.1.1 Asymptotic expansions for potential matrices We assume that the absorption condition A2 holds. Let us also assume that the following perturbation conditions hold for some integer k  0: .k/ ."/ .0/ P18 : pij D pij C eij Œ1; 0" C    C eij Œk; 0"k C o."k / for i; j ¤ 0, where jeij Œr; 0j < 1, r D 1; : : : ; k. .k/ obviously implies that condition T1 (a) holds, i.e., Condition P18 ."/ .0/ ! pij as " ! 0; pij

i ¤ 0;

j 2 X:

(5.1.1)

.0/

Moreover, in the case where k D 0, condition P18 is equivalent to the convergence relation (5.1.1). .k/ in a matrix form. We will rewrite condition P18 For r D 1; : : : ; k, introduce the matrices 2

P."/

."/

p11 6 :: D4 :

."/

::: :: :

."/ 3 p1N :: 7 ; : 5 ."/

pN1 : : : pN N

3 e11 Œr; 0 : : : e1N Œr; 0 7 6 :: :: :: EŒr; 0 D 4 5: : : : eN1 Œr; 0 : : : eN N Œr; 0 2

Here and in the sequel, o."k / denotes a matrix or a vector such that jo."k /j D o."k / as " ! 0. .k/ Condition P18 can be rewritten in the following equivalent matrix form: .k/ : P."/ D P.0/ C EŒ1; 0" C    C EŒk; 0"k C o."k /, where EŒr; 0, r D 1; : : : ; k, P18 are finite matrices.

For j ¤ 0, consider the matrices 2 ."/ ."/ ."/ ."/ p11 : : : p1.j 1/ 0 p1.j C1/ : : : p1N 6 :: :: :: :: :: ."/ D6 jP : : : : 4 : ."/ ."/ ."/ ."/ pN1 : : : pN.j 1/ 0 pN.j C1/ : : : pN N

3 7 7; 5

and also, for r D 1; : : : ; k and j ¤ 0, the matrices 3 2 e11 Œr; 0 : : : e1.j 1/ Œr; 0 0 e1.j C1/ Œr; 0 : : : e1N Œr; 0 7 6 :: :: :: :: :: 5: j EŒr; 0 D 4 : : : : : eN1 Œr; 0 : : : eN.j 1/ Œr; 0 0 eN.j C1/ Œr; 0 : : : eN N Œr; 0

5.1 Asymptotic expansions for absorption and hitting probabilities

261

.k/

Condition P18 yields the following matrix asymptotic expansion: jP

."/

D j P.0/ C j EŒ1; 0" C    C j EŒk; 0"k C o."k /:

(5.1.2)

.0/

It is convenient to set eij Œ0; 0 D pij , i; j ¤ 0, and also EŒ0; 0 D P.0/ and .0/ j EŒ0; 0 D j P , j ¤ 0. As was shown in the proof of Lemma 4.2.1, conditions A2 , T1 (a) and E2 imply .0/ that for "0 > 0 defined in relation (4.2.14) and j 2 X1 , sup j .j P."/ /n j ! 0 as n ! 1:

(5.1.3)

""0

.0/

Relation (5.1.3) implies that for "  "0 and j 2 X1 there exists the inverse matrix ŒI  j P."/ 1 . Denote this matrix by jU

."/

."/

D kj uil k D ŒI  j P."/ 1 :

Elements of the matrix j U."/ have a clear probability meaning, namely, for i; l ¤ 0, j 2 X1.0/ , ."/ j uil

."/

D Ei ıj l ;

(5.1.4)

where ."/

."/

ıj l D

."/

j ^0

X

."/

ı.n1 ; l/ D

nD1

1 X

."/

.j

."/

."/

^ 0 > n  1; n1 D l/:

nD1 ."/

is the number of times the imbedded Markov chain n visits the state l until it first hits one of the states j or 0. .0/ The matrices j U."/ , j 2 X1 can be considered as some kind of potential matrices associated with the matrix P."/ of transition probabilities. The following lemma, which is a direct corollary of Lemma 8.1.2, gives an asymp.0/ totic matrix expansion for the potential matrices j U."/ , j 2 X1 . .k/

Lemma 5.1.1. Let conditions A2 , E2 , and P18 hold. Then .0/

(i) The potential matrices j U."/ have, for every j 2 X1 , the following asymptotic matrix expansions: jU

."/

D j U.0/ C j UŒ1" C    C j UŒk"k C o."k /;

where j UŒr, r D 1; : : : ; k, are finite matrices;

(5.1.5)

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Nonlinearly perturbed semi-Markov processes

(ii) The matrix coefficients j UŒr are given by the following recurrence formulas: .0/ D ŒI  P.0/ 1 and, for r D 1; : : : ; k, j UŒ0 D j U j j UŒr D j UŒ0

r X

j EŒq; 0j UŒr

 q:

(5.1.6)

qD1

Proof. Relations (5.1.2) and (5.1.3) imply that conditions of Lemma 8.1.2 are satisfied by the nonlinearly perturbed matrices I  j P."/ for every j 2 X1.0/ . By applying the statement (iv) of Lemma 8.1.2 to these matrices we prove the claim of Lemma 5.1.1.

5.1.2 Pivotal properties of asymptotic expansions for potential matrices Let us use the following notations for the matrices j UŒr, r D 0; : : : ; k: 3 2 j u11 Œr : : : j u1N Œr 7 6 :: :: :: 5: j UŒr D 4 : : : j uN1 Œr : : : j uN N Œr Explicit expressions for elements of the matrices j UŒr, r D 0; : : : ; k, can be easily derived from the recurrence matrix formulas (5.1.6). We are interested in clarifying pivotal properties of the asymptotic expansions for ."/ the components j uil of elements of the potential matrices j U."/ . A description of such properties is especially important for cyclic elements of these matrices; these are ."/ .0/ elements j uj l for the recurrent-without-absorption states j 2 X1 . For a pair of states .i; j /, where i; j ¤ 0, we define a parameter, called a local asymptotic order, via the coefficients in the asymptotic expansion for the transition ."/ .k/ probabilities pij given in condition P18 , ( r if eij Œn; 0 D 0, n < r but eij Œr; 0 > 0, for r D 0; : : : ; k, r.i; j / D k C 1 if eij Œn; 0 D 0, n  k. Let i; j ¤ 0 and assume that a chain of states .i0 ; i1 ; : : : ; im / is an .i; j /-chain if i0 D i , im D j and i1 ; : : : ; im1 ¤ 0; i; j . For an .i; j /-chain .i0 ; i1 ; : : : ; im /, define a parameter, called a chain asymptotic order, via the corresponding local asymptotic order of subsequent pairs of states in this chain, ( P r if m nD1 r.in1 ; in / D r for r D 0; : : : ; k, r.i0 ; i1 ; : : : ; im / D Pm k C 1 if nD1 r.in1 ; in /  k C 1. We say that an .i; j /-chain .i0 ; i1 ; : : : ; im / is an irreducible chain of states if all the states i1 ; : : : ; im1 are distinct and i1 ; : : : ; im1 ¤ 0; i; j .

5.1 Asymptotic expansions for absorption and hitting probabilities

263

Let .i0 ; : : : ; im / be an .i; j /-chain. If there exists 1  n0 < n00  m  1 such that in0 D in00 , then we say that the .i; j /-chain .i0 ; : : : ; im / contains a loop of states in0 ; : : : ; in00 that starts at the state in0 D in00 . By removing this loop of states, i.e., replacing the subsequence in0 ; : : : ; in00 in the initial .i; j /-chain .i0 ; : : : ; im / with the state in0 , this chain can be replaced with the reduced .i; j /-chain .i0 ; : : : ; in0 ; in00 C1 ; : : : ; im /. It is obvious that (a) the chain asymptotic orders of the initial and the reduced .i; j /-chains are connected by the following inequality: r.i0 ; : : : ; in0 ; in00 C1 ; : : : ; im /  r.i0 ; : : : ; im /:

(5.1.7)

0 Let now .i10 ; : : : ; im 0 / be an .i; j /-chain. We now describe a multi-step reduction algorithm that reduces this .i; j /-chain to an irreducible .i; j /-chain .i1 ; : : : ; im / by removing all loops from the initial .i; j /-chain. 0 At step 0 one should check whether .i10 ; : : : ; im 0 / is an irreducible .i; j /-chain. If 0 / this is so, then the .i; j /-chain .i1 ; : : : ; im / coincides with the .i; j /-chain .i10 ; : : : ; im 0 and the algorithm stops. Otherwise we go step 1. Step 1 starts by finding the index n1 D max.1  n  m0  1 W in0 D i10 /. If n1 D 1, 0 0 then the .i; j /-chain .i10 ; : : : ; im 0 / has no loops of states that start at the state i1 . In this 0 0 case, the .i; j /-chain .i1 ; : : : ; im / coincides with the .i; j /-chain .i1 ; : : : ; im0 /, step 1 of the algorithm stops, and we go to step 2. If n1 > 1, then the loop of states i10 ; : : : ; in0 1 that starts at the state i10 D in0 1 should be removed from the .i; j /-chain 0 /. This means that this .i; j /-chain should be replaced with the reduced .i10 ; : : : ; im 0 0 .i; j /-chain .i00 ; i10 ; in0 1 C1 ; : : : ; im 0 /. If the .i; j /-chain is irreducible, then the .i; j /0 chain .i1 ; : : : ; im / coincides with the .i; j /-chain .i00 ; i10 ; in0 1 C1 ; : : : ; im 0 / and the algorithm stops. Otherwise we start step 2. Step 2 starts by finding the index n2 D max.n1  n  m0  1 W in0 D in0 1 C1 /. 0 If n2 D n1 C 1, then the .i; j /-chain .i10 ; : : : ; im 0 / has no loops of states that start 0 at the state in1 C1 . In this case, the .i; j /-chain .i1 ; : : : ; im / coincides with the .i; j /0 / step 2 of the algorithm ends and we pass to step 3. If chain .i00 ; i10 ; in0 1 C1 ; : : : ; im 0 n2 > n1 C1, then the loop of states in0 1 C1 ; : : : ; in0 1 that starts at the state in0 1 C1 ; : : : ; in0 2 0 should be removed from the .i; j /-chain .i00 ; i10 ; in0 1 C1 ; : : : ; im 0 /. This means that this 0 0 0 0 .i; j /-chain is replaced with the reduced .i; j /-chain .i0 ; i1 ; in1 C1 ; in0 2 C1 ; : : : ; im 0 /. If this reduced .i; j /-chain is irreducible, then the .i; j /-chain .i1 ; : : : ; im / coincides 0 with the .i; j /-chain .i00 ; i10 ; in0 1 C1 ; in0 2 C1 ; : : : ; im 0 / and the algorithm stops. Otherwise, the algorithm continues. The algorithm stops when the subsequent removing of loops as described above results in an irreducible .i; j /-chain .i1 ; : : : ; im /. It is obvious that the final irreducible .i; j /-chain .i1 ; : : : ; im / is uniquely deter0 mined by the initial .i; j /-chain .i10 ; : : : ; im 0 /. Also, inequality (5.1.7) implies that 0 r.i0 ; : : : ; im /  r.i00 ; : : : ; im 0 /:

(5.1.8)

264

5

Nonlinearly perturbed semi-Markov processes

 Denote by Rij the set of all .i; j /-chains and by Rij the set of all irreducible .i; j /chains. The multi-step reduction algorithm described above defines a mapping of the set  Rij to the set Rij . Let Rij .i0 ; : : : ; im / be the set of inverse images of an irreducible .i; j /-chain .i0 ; : : : ; im / with respect to this mapping. By the definition, (c) the sets Rij .i0 ; : : : ; im / have empty intersections for different irreducible .i; j /-chains and [  Rij D Rij .i0 ; : : : ; im /: (5.1.9) .i0 ;:::;im /2Rij

We will refer to the .i; j /-chain .i0 ; : : : ; im / as the core .i; j /-chain for the set of .i; j /-chains Rij .i0 ; : : : ; im /. The multi-step reduction algorithm described above allows to also define, for the set of .i; j /-chains Rij .i0 ; : : : ; im /, intermediate sets of .i; j /-chains obtained after performing a fixed number of steps of this algorithm. .1/ .i0 ; : : : ; im / the set of all .i; j /-chains obtained after the first Denote by Rij step of the multi-step reduction algorithm applied to the .i; j /-chains from the set .2/ Rij .i0 ; : : : ; im /, by Rij .i0 ; : : : ; im / the set of all .i; j /-chains obtained after the second step of the multi-step reduction algorithm applied to the .i; j /-chains from .1/ .m1/ the set Rij .i0 ; : : : ; im /, etc. Finally, denote by Rij .i0 ; : : : ; im / the set of all .i; j /-chains obtained after step .m  1/ of the multi-step reduction algorithm ap.m2/ .i0 ; : : : ; im /. It is readily seen that the set plied to .i; j /-chains in the set Rij .m1/

Rij

.i0 ; : : : ; im / D f.i0 ; : : : ; im /g contains only the core .i; j /-chain .i0 ; : : : ; im /.

.r / .i0 ; : : : ; im / satisfy the following inclusion relations: By the definition, the sets Rij .1/

.m1/

Rij .i0 ; : : : ; im / Rij .i0 ; : : : ; im /    Rij

.i0 ; : : : ; im /:

(5.1.10)

Finally, let us define a global asymptotic order of a pair of states .i; j /, where i; j ¤ 0, via the chain asymptotic orders of the .i; j /-chains, r.i; O j/ D

min

0  .i00 ;1 ;:::;im 0 /2Rij

0 r.i00 ; : : : ; im 0 /:

It follows from the remarks above that (d) the global asymptotic order can also be defined in a simpler way, r.i; O j/ D

min

.i0 ;i1 ;:::;im /2Rij

r.i0 ; i1 ; : : : ; im /:

(5.1.11)

Indeed, the inequalities for the chain orders of .i; j /-chains and their reduced counterparts given in (5.1.7) and (5.1.8) imply that (e) the following relation holds for any irreducible .i; j /-chain .i0 ; i1 ; : : : ; im /: r.i0 ; i1 ; : : : ; im / D

min

0 .i00 ;:::;im 0 /2Rij .i0 ;i1 ;:::;im /

0 r.i00 ; : : : ; im 0 /:

(5.1.12)

5.1 Asymptotic expansions for absorption and hitting probabilities

265

Relation ( 5.1.12) implies that r.i; O j/ D D D

min

0 r.i00 ; : : : ; im 0/

min

min

0  .i00 ;1 ;:::;im 0 /2Rij

0 .i0 ;i1 ;:::;im /2Rij .i00 ;:::;im 0 /2Rij .i0 ;i1 ;:::;im /

min

.i0 ;i1 ;:::;im /2Rij

0 r.i00 ; : : : ; im 0/

r.i0 ; i1 ; : : : ; im /:

(5.1.13)

Formula (5.1.11) has an obvious advantage as compared with formula (5.1.2). In deed, the set Rij is infinite, while Rij is a finite set, since any irreducible .i; j /-chain .i0 ; i1 ; : : : ; im / has the length m  N . Thus, a calculation of the global asymptotic order rij according to formula (5.1.11) requires a finite number of comparison operations. It useful to note that in application models the transition matrices P."/ usually have a number of zero elements, (f) pl."/ 0 l 00 D 0 for "  0. A pair .l 0 ; l 00 / such that (f) holds obviously has the local asymptotic order equal to k C 1. This implies that (g) an irreducible .i; j /-chain .i0 ; i1 ; : : : ; im / has the chain asymptotic order r.i0 ; i1 ; : : : ; im / D k C 1 if it includes a pair of states .in1 ; in / D .l 0 ; l 00 / which satisfies (a). 0 Denote by Rij the set of all irreducible .i; j /-chains .i0 ; i1 ; : : : ; im / that do not include pairs of states satisfying (a). In follows from the remarks above that, (h) in order to calculate the global asymptotic order of a pair of states .i; j /, it is actually necessary to calculate and to compare 0 the chain asymptotic orders only for irreducible .i; j /-chains from set Rij , i.e., r.i; O j/ D

min

0

.i0 ;i1 ;:::;im /2Rij

r.i0 ; i1 ; : : : ; im /:

(5.1.14)

In many cases, this simplifies the calculation of the global asymptotic order of a pair of states .i; j /. Another useful remark is that (i) a .i; j /-chain .i0 ; i1 ; : : : ; im / has r.i0 ; i1 ; : : : ; im / .0/ .0/ D 0 if and only if pi0 i1    pim1 im > 0. It is also useful to note that, according to inequality (5.1.7), (j) if an .i; j /-chain has the chain asymptotic order equal to 0 and belongs to the class Rij .i0 ; i1 ; : : : ; im / with a core irreducible chain .i0 ; i1 ; : : : ; im /, then this core irreducible chain also has the chain asymptotic order r.i0 ; i1 ; : : : ; im / D 0. These remarks show that (k) a pair .i; j / has the global asymptotic order rO .i; j / D 0 if and only if there exists an irreducible .i; j /-chain .i0 ; i1 ; : : : ; im / with the chain .0/ .0/ asymptotic order r.i0 ; i1 ; : : : ; im / D 0 or, equivalently, with pi0 i1    pim1 im > 0. The notion of the global asymptotic order allows to give a constructive description ."/ of pivotal properties of the asymptotic expansion of the cyclic elements j uj l for j 2 .0/

X1 ; l ¤ 0.

266

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Nonlinearly perturbed semi-Markov processes

Introduce the sets ( fl 2 X0 W j uj l Œn D 0; n D 0; : : : ; r  1; j uj l Œr ¤ 0g Zj;r D fl 2 X0 W j uj l Œn D 0; n D 0; : : : ; kg

for r D 1; : : : ; k, for r D k C 1.

We shall show below that the sets Zj;r D Zr , r D 0; : : : ; k C 1, do not depend on .0/ the choice of the state j 2 X1 . Also, by the definition, the sets Zr , r D 0; : : : ; k C 1, are disjoint, i.e., Zr 0 \ Zr 00 D ¿;

r 0 ; r 00 ¤ 0;

r 0 ¤ r 00 :

(5.1.15)

Note also that, by the definition, some of the sets Zr , r D 1; : : : ; k C 1, can be S .0/ empty, but, as we shall show, Z0 D X1.0/ and kC1 rD1 Zr D X2 . The following lemma describes properties of the sets Zr , r D 0; : : : ; k C 1, mentioned above, which permits to not only prove invariance properties but also to give a constructive description of these sets in terms of the corresponding global asymptotic orders. .k/ Lemma 5.1.2. Let conditions A2 , E2 , and P18 hold. Then

(i) The global asymptotic order r.j; O l/ D r.l/ O does not depend on the choice of the .0/ state j 2 X1 for every state l ¤ 0. .0/

(ii) Zj;r D Zr , r D 0; : : : ; k C1, do not depend on the choice of the state j 2 X1 . (iii) Z0 D X1.0/ ,

SkC1 r D1

Zr D X2.0/ .

(iv) Zr D fl ¤ 0 W r.l/ O D rg, r D 0; : : : ; k C 1. .0/

Proof. Let us prove statement (i) of the lemma. Note first of all that X1 ¤ ¿ by .0/ condition E2 . Statement (i) is trivial if the set X1 contains only one state. So, assume that it contains at least two states. .0/ 0 00 00 Let j 0 ; j 00 2 X1 , j 0 ¤ j 00 and l ¤ 0. Let also .i00 ; : : : ; im 0 / and .i0 ; : : : ; im00 / 0 00 be irreducible .j ; l/-chain and .j ; l/-chain which have the chain asymptotic orders equal to the global asymptotic orders r.j O 0 ; l/ and rO .j 00 ; l/, respectively. 0 00 O ; l/ and show that this assumption implies that (m) Assume that (l1 ) rO .j ; l/ < r.j one can find an irreducible .j 00 ; l/-chain with the chain asymptotic orderless than or equal to r.j O 0 ; l/. There are two cases to be considered. The first case is where the state j 00 coin0 0 0 cides with some state in0 in the .j 0 ; l/-chain .i00 ; : : : ; im 0 /. Then, .in ; : : : ; im0 / is an 00 0 0 0 0 0 irreducible .j ; l/-chain and r.in ; : : : ; im0 /  r.i0 ; : : : ; im0 / D rO .j ; l/. The second case is where the state j 00 does not coincide with any state in0 in the 0 0 /. Denote by V the set of all states i 0 from the .j 0 ; l/-chain .j ; l/-chain .i00 ; : : : ; im 0 n .0/ 0 / that belong to the set X . We use the notations introduced in Section .i00 ; : : : ; im 0 1

267

5.1 Asymptotic expansions for absorption and hitting probabilities .0/

.0/

.0/

.0/

4.2 and consider the hitting probabilities 0 fj 00 Zi 0 D Pj 00 fV < 0 ;  .0/ .V / D in0 g, n

in0 2 V . Since X1.0/ is the set of all recurrent-without-absorption states, there ex.0/ ists at least one state in0 2 V such that 0 fj 00 V i 0 > 0. Obviously, this probability is n positive if and only if (n1 ) there exists an irreducible .j 00 ; in /-chain .i0 ; : : : ; im1 /    pi.0/ > 0. The transition probasuch that i1 ; : : : ; im 62 V [ f0g and pi.0/ 0 i1 m1 im .0/ D 0 for all states i 2 X1.0/ ; j 2 X2.0/ (if X2.0/ ¤ ¿). Thus, bilities are pij

(n2 ) all states satisfy i0 ; : : : ; im1 2 X1.0/ . It follows from (n1 ) and (n2 ) that the 0 chain .i1 ; : : : ; im ; in0 ; : : : ; im 0 / is irreducible. Obviously r.i1 ; : : : ; im / D 0 and the 0 0 0 0 0 chain order r.in ; : : : ; im0 /  r.i10 ; : : : ; im 0 /. Therefore, r.i1 ; : : : ; im ; in ; : : : im0 /  0 0 0 0 0 r.i1 ; : : : ; im / C r.in ; : : : ; im0 /  r.i1 ; : : : ; im0 / D r.j O ; l/. Statement (m) contradicts (l1 ), since the global asymptotic order rO .j 00 ; l/, by the definition, is greater than or equal to the chain asymptotic order for any irreducible .j 00 ; l/-chain. Thus the inequality (l1 ) is impossible and, therefore, (h2 ) rO .j 00 ; l/  r.j O 0 ; l/. In the O 0 ; l/  r.j O 00 ; l/. Inequalities (l2 ) and (l3 ) imply same way, one can prove that (l3 ) r.j 0 00 that r.j O ; l/ D r.j O ; l/, which proves (m) in the lemma. In what follows, we will use two formulas. The first one is the obvious formula ."/ j uij

."/ D Ei ıjj D ı.i; j /;

i; j ¤ 0;

(5.1.16)

where ı.i; j / is the Kronecker delta-function. The second one is a formula that plays a key role in the further proof, ."/

."/

."/

."/

j uil D Ei ıj l D 0jfil El ıj l D

."/ 0jfil ."/

1  0jfl l

; i; j; l ¤ 0; j ¤ l;

(5.1.17)

We prove statements (ii)–(iii) in a simple case where condition E02 holds. Recall that .0/ condition E02 means that the set of recurrent-without-absorption states is X1 D X0 , where X0 D f1; : : : ; N g. The case N D 1 is trivial. Indeed, by formula (5.1.16), the expectation (o1 ) satisfies .0/ u j jj D 1. Thus, Z0 D fj g. This automatically implies that Z1 ; : : : ; ZkC1 D ¿. In this case, the only irreducible .j; j /-chain is the chain .j; j / of length 1. The state .0/ > 0. Thus, (p1 ) the chain j is recurrent-without absorption, which implies that pjj asymptotic order is r.j; j / D 0 and, therefore, the global asymptotic order satisfies rO .j / D r.j; O j / D 0. The remarks above imply that Z0 D fj g D fj W rO .j / D 0g. .0/ Let us now assume that N > 1. Formula (5.1.16) implies that (o2 ) j ujj D 1. Also, .0/

.0/

.0/

in this case, 0jfj l > 0 and 0jfl l < 1 for all j; l 2 X1 , j ¤ l. Therefore, (o3 ) the .0/

.0/

expectations satisfy Ej ıj l > 0 for all j; l 2 X1 ; j ¤ l. Thus the set Zj;0 D Z0 .0/

does not depend on j 2 X1

.0/

and, moreover, it coincides with the set X1

D X0 .

268

5

Nonlinearly perturbed semi-Markov processes .0/

This automatically implies that Z1 ; : : : ; ZkC1 D ¿. Since all j 2 X1 are recurrent.0/ .0/ without-absorption states, the probabilities are positive, 0jfj l > 0 for all j; l 2 X1 . .0/

We have 0jfj l

> 0 if and only if there exists an irreducible .i; l/-chain .i0 ; : : : ; im /

.0/ pi0 i1

.0/

   pim1 im > 0 or, equivalently, r.i0 ; : : : ; im / D 0. This implies such that that (p2 ) the global asymptotic order is zero, rO .j; l/ D rO .l/ D 0. This holds for all l 2 X1.0/ . Thus the set fl ¤ 0 W r.l/ O D 0g coincides with the set X1.0/ D Z0 . Let us now consider the case where condition E002 is satisfied. Recall that condi.0/ tion E02 means that X1 ¤ ¿ but also the set of non-recurrent-without-absorption is .0/ nonempty, X2 ¤ ¿. The proofs of the claims that (o4 ) all states l 2 X1.0/ belong to the set Zj;0 for every j 2 X1.0/ , and that (p3 ) the global asymptotic order is zero, r.j; O l/ D r.l/ O D 0, for .0/ .0/ all states l 2 X1 repeats the above. Let us prove that (o5 ) all states l 2 X2 do not .0/ .0/ belong to the set Zj;0 for every j 2 X1 . Indeed, by the definition of the set X1 , the .0/ .0/ .0/ probabilities pj l D 0 for all states j 2 X1 , l 2 X2 . This implies, in an obvious .0/ .0/ D 0. Also, by condition E002 , we have that 0jfl l < 1 for all j 2 0jfj l .0/ .0/ .0/ X1 ; l 2 X2 . Therefore, by formula (5.1.17), the expectations are zero, Ej ıj l D 0, .0/ .0/ for all j 2 X1 , l 2 X2 . Obviously, (o4 ) and (o5 ) imply that the set Zj;0 D Z0 .0/ .0/ does not depend on j 2 X1 and, moreover, this set coincides with the set X1 . Let O l/ D r.l/ O > 0, for us also prove that (p4 ) the global asymptotic order is positive, r.j; all states l 2 X2.0/ . Indeed, as was mentioned above, in this case, 0jfj.0/ D 0 for all l .0/ .0/ .0/ j 2 X1 , l 2 X2 . Now, 0jfj l D 0 if and only if there are no irreducible .j; l/-

way, that

chains with the chain asymptotic order 0. This implies that the global asymptotic order .0/ .0/ satisfies r.j; O l/ D r.l/ O > 0. This holds for all j 2 X1 ; l 2 X2 . Obviously, (p3 ) .0/ and (p4 ) imply that the set fl ¤ 0 W rO .l/ D 0g coincides with the set X1 D Z0 . Note SkC1 that this also implies that r D1 Zr D X2.0/ . Let us now prove that (q) Zj;r D fl ¤ 0 W r.l/ O D rg for all states j 2 X1.0/ and r D 1; : : : ; k C 1. This will complete the proof of the lemma. .k/ As was mentioned above, condition P18 implies that condition T1 (a) is satisfied, and, as was shown in the proof of Lemma 4.2.1, conditions A2 , E2 , and T1 (a) imply that ."/

.0/

.0/

Ei ıj l ! Ei ıj l < 1 as " ! 0; i; l ¤ 0; j 2 X1 : .0/

.0/

(5.1.18)

.0/

As was mentioned above, 0jfl l < 1 for all j 2 X1 , l 2 X2 . Thus,  1; El ıj.0/ l

j 2 X1.0/ ;

l 2 X2.0/ : ."/

(5.1.19) ."/

.0/

.0/

According to Lemma 5.1.1, the expectations j uj l D Ej ıj l , j 2 X1 , l 2 X2 admit asymptotic expansions of order k. Hence, formula (5.1.17) used in the case

269

5.1 Asymptotic expansions for absorption and hitting probabilities .0/

.0/

i D j 2 X1 ; l 2 X2 , relations (5.1.18) and (5.1.19) imply that, (r) for every .0/ j 2 X1 and r D 1; : : : ; k C 1, we will have l 2 Zj;r if and only if there exists a ."/ coefficient fj l Œr > 0 such that the probability 0jfj l has the following asymptotic representation: ( "r fj l Œr C o."r / if r D 1; : : : ; k, ."/ (5.1.20) 0jfj l D if r D k C 1. o."k / ."/

The probability 0jfj l can be represented as the following sum: X ."/ ."/ pi."/ 0 0 p 0 0jfj l D i i0 i X

X

D

m0 1 m0

0 1

0  .i00 ;:::;im 0 /2Ri l

."/

0 .i0 ;:::;im /2Ri l .i00 ;:::;im 0 /2Ri l .i0 ;:::;im /

."/

pi 0 i 0    pi 0

i0 m0 1 m0

0 1

(5.1.21)

:

0 / the number of states i in the chain .i 0 ; : : : ; i 0 / from Denote by Nq .i00 ; : : : ; im 0 q 0 m0 .q1/ .i0 ; : : : ; im / for q D 1; : : : ; m  1. Here we set, for convenience, the set Rj l .0/

Rj l .i0 ; : : : ; im / D Rj l .i0 ; : : : ; im /. Let us also denote Vq D Vq .i0 ; : : : ; iq1 ; im / D f0; j; l; i1 ; : : : ; iq1 g for q D ."/ ."/ ."/ ."/ 1; : : : ; m  1, and fq .i0 ; : : : ; iq1 ; im / D Vqfiq iq D Piq fiq < Vq g for q D 1; : : : ; m  1. A key role in our calculations plays the following formulas which hold for every q D 1; : : : ; m  1: X ."/ pi."/ 0 0 p 0 i i0 i .q1/

0 .i00 ;:::;im 0 /2Rj l

D

.i0 ;:::;im /

m0 1 m0

0 1

1 X

X

nD1 .i 0 ;:::;i 0 /2R.q1/ .i ;:::;i /; N .i 0 ;:::;i 0 /Dn 0 m q 0 0 jl m0 m0

D

1 X

i0 m0 1 m0

0 1

.fq."/ .i0 ; : : : ; iq1 ; im //n1

nD1

X



.q/

00 .i000 ;:::;im 00 /2Rj l .i0 ;:::;im /

D

."/ pi."/ 0 0 p 0 i i

."/

."/

pi 00 i 00    pi 00 0 1

i 00 m00 1 m00

1 1

fq."/ .i0 ; : : : ; iq1 ; im / 

X

.q/ 00 .i000 ;:::;im 00 /2Rj l .i0 ;:::;im /

."/ pi."/ 00 00    p 00 i i 0 1

i 00 m00 1 m00

:

(5.1.22)

270

5

Nonlinearly perturbed semi-Markov processes

Combining formulas (5.1.21)–(5.1.22) together and taking into account that the set contains the only core irreducible .j; l/-chain .i0 ; : : : ; im /, we get the following important formula:

.i0 ; : : : ; im / Rj.m1/ l ."/ 0jfj l

D

X

m1 Y

1 1

.i0 ;:::;im /2Rj l qD1

."/

fq."/ .i0 ; : : : ; iq1 ; im /

."/

 pi0 i1    pim1 im : (5.1.23)

By Lemma 4.2.5, for every q D 1; : : : ; m  1, fq."/ .i0 ; : : : ; iq1 ; im / D

."/ Vqfiq iq

! fq.0/ .i0 ; : : : ; iq1 ; im / D

.0/ Vqfiq iq

as " ! 0:

.0/

(5.1.24) .0/

Since j is a recurrent-without-absorption state, fq .i0 ; : : : ; iq1 ; im / D Vqfiq iq < 1, q D 1; : : : ; m  1, and, therefore, the following relation holds: m1 Y qD1

1 1

.0/ fq .i0 ; : : : ; iq1 ; im /

> 0:

(5.1.25)

It follows from relations (5.1.23), (5.1.24), and (5.1.25) that (s) the asymptotic relation (5.1.20) holds if and only if the global asymptotic order satisfies the following relation min.i0 ;:::;im /2Rj l r.i0 ; : : : ; im / D rO .j; l/ D r for some r D 1; : : : ; k C 1. Moreover, in the cases r D 1; : : : ; k, the coefficients fj l Œr can be calculated with the use of the following formula: m1 Y X 1 fj l Œr D .0/ .i0 ;:::;im /2Rj l ; rj l .i0 ;:::;im /Dr qD1 1  fq .i0 ; : : : ; iq1 ; im / 

n Y

ein1 in Œr.in1 ; in /; 0:

(5.1.26)

nD1

O l/ D rg for every It follows from propositions (r) and (s) that Zj;r D fl ¤ 0 W r.j; r D 1; : : : ; k C1. According to claim (i) in the lemma, the asymptotic orders r.j; O l/ D .0/ rO .l/, l ¤ 0, do not depend on j 2 X1 . Thus, (t) the set Zj;r D fl ¤ 0 W r.j; O l/ D rg O D rg for every r D 1; : : : ; k C 1. coincides with the set Zr D fl ¤ 0 W r.l/ The proof of the lemma is complete.

5.1.3 Asymptotic expansions for solutions of hitting type systems of linear equations ."/

Let us consider functions yi , i ¤ 0, and assume that they satisfy the following nonnegativity condition: ."/

S1 : yi

 0 for "  0 and i ¤ 0;

271

5.1 Asymptotic expansions for absorption and hitting probabilities

and that the following perturbation condition holds: .k/

."/

.0/

P19 : yi D yi C yi Œ1" C    C yi Œk"k C o."k / for i ¤ 0, where jyi Œrj < 1, r D 1; : : : ; k. For r D 1; : : : ; k, consider the vectors 2 ."/ 3 y1 7 6 y."/ D 4 ::: 5 ; ."/

yN

3 y1 Œr 7 6 :: yŒr D 4 5: : yN Œr 2

.k/

Condition P19 can be rewritten in the following equivalent vector form: .k/

P19 : y."/ D y.0/ C yŒ1" C    C yŒk"k C o."k /, where yŒr, r D 1; : : : ; k, are finite vectors. .0/

It is convenient to define yi Œ0 D yi , i ¤ 0, and yŒ0 D y.0/ . .0/ For every j 2 X1 , consider the system of linear equations D y."/ C j P."/ x."/ x."/ j j :

(5.1.27)

Under conditions A2 , E2 , and T1 (a), system (5.1.27) has a unique solution for .0/ every "  "0 and j 2 X1 , ."/ 1 ."/ x."/ j D ŒI  j P  y :

(5.1.28) ."/

It is also useful to note that if condition S1 is satisfied, the vector xj has all components nonnegative. The following lemma gives vector asymptotic expansions for solutions of these systems. .k/

.k/

Lemma 5.1.3. Let conditions A2 , E2 , P18 , S1 , and P19 hold. Then ."/

(i) The vector xj

.0/

has the following asymptotic expansion for every j 2 X1 : .0/ x."/ C xj Œ1" C    C xj Œk"k C o."k /; j D xj

(5.1.29)

where xj Œr, r D 1; : : : ; k, are finite vectors. .0/

(ii) The vector coefficients xj Œr are given by the recurrence formulas xj Œ0 D xj .0/ D ŒI  P.0/ 1 y.0/ and, for r D 0; : : : ; k, j UŒ0y j xj Œr D j UŒ0.yŒr C

r X qD1

j EŒq; 0 xj Œr

 q/:

D

(5.1.30)

272

5

Nonlinearly perturbed semi-Markov processes

(iii) The vector coefficients xj Œr can also be calculated by the following formulas for r D 0; : : : ; k: xj Œr D

r X

j UŒp yŒr

 p:

(5.1.31)

pD0 .k/ , expressed in the vector form, Proof. Relations (5.1.2) and (5.1.3) and condition P19 imply that conditions of Lemma 8.1.4 are satisfied for the system of nonlinearly per.0/ turbed linear equations (5.1.27) for every j 2 X1 . By applying this lemma to systems (5.1.27), we get the claims of Lemma 5.1.1. .0/

For r D 1; : : : ; k and j 2 X1

denote

2

."/

xj

3 ."/ x1j 6 : 7 : 7 D6 4 : 5; ."/ xNj

3 x1j Œr 7 6 :: xj Œr D 4 5: : xNj Œr 2

Explicit expressions for components of the vectors xj Œr, r D 0; : : : ; k, can be derived from recurrence formulas (5.1.30) or formulas (5.1.31). There is a simple probabilistic interpretation of solutions of system (5.1.27). Indeed, using the representation for elements of the matrix ŒI  j P."/ 1 given by formula (5.1.4), we can write for every i ¤ 0, j 2 X1.0/ the following representation for ."/ components of the vector xj , ."/

xij D

X l¤0

."/ ."/ j uil yl

D

X

."/

."/ ."/

Ei ıj l yl

."/

j ^0

D Ei

X

nD1

l¤0

y

."/ ."/

n

(5.1.32)

:

Formula (5.1.32) permits to refer to systems of type (5.1.27) as a hitting system of linear equations. Hitting systems of type (5.1.27) play an important role in our analysis. Hitting and absorption probabilities and power type moments for hitting times are solutions of such systems. We are especially interested in a study of pivotal properties of asymptotic expan."/ sions for the components xij of the solutions x."/ of system (5.1.32). A description j ."/

of such properties is especially important for the cyclic components xjj of the vectors ."/

xj .

.k/

Let us use perturbation condition P19 to classify the states i ¤ 0 by the order of the first nonzero coefficient in the corresponding asymptotic expansion.

273

5.1 Asymptotic expansions for absorption and hitting probabilities

Consider the sets ( Yr D

fi 2 X0 W yi Œn D 0; n D 0; : : : ; r  1; yi Œr ¤ 0g fi 2 X0 W yi Œn D 0; n D 0; : : : ; kg

for r D 1; : : : ; k, for r D k C 1,

where X0 D f1; : : : ; N g. Note that, by the definition, (a) some of the sets Yr , r D 0; : : : ; k C 1, could be S empty, and (b) Yr 0 \ Yr 00 D ¿, r 0 ¤ r 00 , but (c) kC1 rD0 Yr D X0 . We are now in a position to give a clear description of pivotal properties of asymp."/ .0/ totic expansions for the cyclic components xjj , j 2 X1 . Define the sets Wr D

[

.Yr 0 \ Zr 00 /; r D 0; : : : ; k;

WkC1 D

r 0 Cr 00 Dr

[

.Yr 0 \ Zr 00 /:

r 0 Cr 00 kC1

By the definition, (d) some of the sets Wr , r D 0; : : : ; k C 1, could be empty, (e) S Wr 0 \ Wr 00 D ¿; r 0 ¤ r 00 , but (f) kC1 r D0 Wr D X0 . .0/

The case where condition E02 holds, is a simple one. Indeed, here Z0 D X1 and, therefore, Z1 ; : : : ; ZkC1 D ¿. Thus, Wr D Y r ;

D X0 ,

r D 0; : : : ; k C 1:

For an integer 0  h  k, consider the following two conditions: .h/

O1 : Wr D ¿, for r  h; and O2.h/ : Wr D ¿, for r < h, but Wh ¤ ¿. .k/

.k/

Lemma 5.1.4. Let conditions A2 , E2 , P18 , S1 , and P19 hold. Then (i) If condition O1.h/ holds for some 0  h  k, then xjj Œn D 0, n D 0; : : : ; h, j 2 X1.0/ . .h/

(ii) If condition O2 holds for some 0  h  k, then xjj Œn D 0, n D 0; : : : ; h  1, .0/ .0/ j 2 X1 , but xjj Œh > 0, j 2 X1 . Proof. By using asymptotic expansions (5.1.5) for the potential matrices j U."/ , and .k/ ."/ condition P19 , relation (5.1.32) can be rewritten for the cyclic components xjj , j 2

274

5

Nonlinearly perturbed semi-Markov processes

.0/

X1 , in the following form: X ."/ ."/ ."/ xjj D j uil yl l¤0

D

X

X

X

0r k r 0 Cr 00 Dr l2Yr 0 \Zr 00

D

X

X

0r k

r 0 Cr 00 Dr

0

 @

X

X

."/ ."/ j uil yl

0 @

l2Yr 0 \Zr 00

1

X

X

C

."/ ."/ j uil yl

r 0 Cr 00 kC1 l2Yr 0 \Zr 00

X

1

"p

00

j uj l Œp

00

A

r 00 p 00 k

p0

" yl Œp 0 A C o."k /

r 0 p 0 k

D

X 

 "r Lj Œr; r C    C "k Lj Œk; r C o."k /

0r k

D

X

"r Lj Œr; 0 C    C Lj Œr; r C o."k /;

(5.1.33)

0r k

where the coefficients Lj Œp; r, p D r; : : : ; k, r D 0; : : : ; k, are readily calculated by multiplying the inner sums in (5.1.33) and collecting the coefficients at "p . For every r D 0; : : : ; k, (g) the coefficients Lj Œp; r, p D r; : : : ; k, are sums of numbers from the set Wr . In particular, (h) the coefficients Lj Œr; r have the following form for r D 0; : : : ; k: X X 00 0 Lj Œr; r D (5.1.34) j uj l Œr  yl Œr : r 0 Cr 00 Dr l2Yr 0 \Zr 00

Formula (5.1.33) implies that (i) the coefficients xjj Œr can be represented in the following form for r D 0; : : : ; k: X Lj Œr; p: (5.1.35) xjj Œr D 0pr .h/

If condition O1 holds for some 0  h  k, then Wr D ¿, r  h. Thus, (j) the sum that defines the coefficients LŒp; r, p D r; : : : ; k, are actually sums over the empty set for r  h. Therefore, (k) LŒp; r D 0, p D r; : : : ; k, for r  h. The claim (k) and formula (5.1.35) imply that (l) xjj Œr D 0, r  h. This proves the first statement of the lemma. If condition O2.h/ holds for some 0  h  k, then Wr D ¿, r  h  1, but Wh ¤ ¿. This means that condition O1.h1/ holds and, therefore, (m) the coefficients satisfy xjj Œr D 0; r  h  1, as was proved above. Moreover, similarly to the above, (n) LŒp; r D 0, p D r; : : : ; k for r  h  1. Obviously, (n) and formula (5.1.35) imply that, in the case under consideration, (o) xjj Œh D Lj Œr; r.

275

5.1 Asymptotic expansions for absorption and hitting probabilities

If Wh ¤ ¿, then Yr 0 \ Zr 00 ¤ ¿ for some non-negative integer r 0 and r 00 such that r 0 C r 00 D h. By the definition of the set Zr 00 , we have j uj l Œr 00  > 0, l 2 Zr 00 . Also, by the definitionP of the set Yr 0 and condition S1 , the coefficients satisfy yl Œr 0  > 0, l 2 Yr 0 . Thus, l2Yr 0 \Zr 00 j uj l Œr 00 yl Œr 0  > 0. Therefore, xjj Œr D Lj Œr; r > 0 according to formula (5.1.35). This proves the second statement of the lemma.

5.1.4 Asymptotic expansions for absorption and hitting probabilities .k/ It is useful to note that condition P18 automatically implies that a similar asymptotic ."/ expansion takes place for the local absorption probabilities pi0 , i ¤ 0,

X

."/

pi0 D 1 

."/

pij

j ¤0

X

D1

.0/

.pij C eij Œ1; 0" C    C eij Œk; 0"k C o."k //

j ¤0

D

.0/ pi0

C ei0 Œ1; 0" C    C ei0 Œk; 0"k C o."k /;

where, for r D 1; : : : ; k, ei0 Œr; 0 D 

X

(5.1.36)

eij Œr; 0:

j ¤0

For r D 1; : : : ; k and j 2 X, we introduce the vectors, 2 3 3 2 ."/ p1j e1j Œr; 0 6 : 7 7 6 :: 6 :: 7 ; ej Œr; 0 D 4 p."/ 5: : j D4 5 ."/ eNj Œr; 0 p Nj

.k/

Condition P18 and asymptotic relation (5.1.36) give the following vector asymptotic expansion for j 2 X: .0/ k k p."/ j D pj C ej Œ1; 0" C    C ej Œr; 0" C o." /:

(5.1.37)

.0/ , i; j ¤ 0. It will be convenient in the sequel to set Recall that eij Œ0; 0 D pij

ej Œ0; 0 D p.0/ j , j ¤ 0. Now, let us recall that the absorption probabilities satisfy ."/ jfi0

."/

."/

i; j ¤ 0;

."/

i; j ¤ 0:

D Pi f0 < j g;

and, for the hitting probabilities, we have ."/ 0 fij

."/

D Pi fj

< 0 g;

276

5

Nonlinearly perturbed semi-Markov processes

For j ¤ 0, consider the vectors 2 ."/ 3 jf10 6 :: 7 ."/ j f0 D 4 : 5;

2 ."/ 0 fj

."/ jfN 0

."/ 0 f1j

6 D6 4

3

:: 7 : 7 5: ."/ 0 fNj

As was shown in Lemma 4.2.1, conditions A2 , T1 (a), and E2 imply for every .0/ "  "0 and j 2 X1 that the absorption probabilities uniquely solve the following system of linear equations: ."/ j f0

."/

."/

D p0 C j P."/ j f0 ;

(5.1.38)

.0/

and that for every "  "0 and j 2 X1 , the hitting probabilities make a unique solution the system ."/ 0 fj

."/

D pj

."/

C j P."/ 0 fj :

(5.1.39)

The following lemma is a direct corollary of Lemma 5.1.3 applied to the absorption ."/ .0/ probability vectors j f0 ; j 2 X1 . .k/

Lemma 5.1.5. Let conditions A2 , E2 , and P18 , hold. Then .0/

."/

(i) For every j 2 X1 , the vector j f0 has the following asymptotic expansion: ."/ j f0

.0/

D j f0 C j b0 Œ1; 0" C    C j b0 Œk; 0"k C o."k /;

(5.1.40)

where j b0 Œr; 0, r D 1; : : : ; k, are finite vectors. (ii) The vector coefficients j b0 Œr; 0 are given by the recurrence formulas j b0 Œ0; 0 D .0/ .0/ .0/ 1 .0/ j f0 D j UŒ0p0 D ŒI  j P  p0 and, for r D 0; : : : ; k,   r X (5.1.41) j EŒq; 0 j b0 Œr  q; 0 : j b0 Œr; 0 D j UŒ0 e0 Œr; 0 C qD1

(iii) The vector coefficients j b0 Œr; 0 can also be calculated by the following formulas for r D 0; : : : ; k: j b0 Œr; 0

D

r X

j UŒp e0 Œr

 p; 0:

(5.1.42)

pD0

Proof. Lemma 5.1.3 can be applied to the system of linear equations (5.1.38) for every ."/ j 2 X1.0/ . In this case, the vector p."/ 0 plays the role of the vector y . The nonnegativity condition S1 obviously holds. Also, the asymptotic relation (5.1.37) implies .k/ that condition P19 holds for j D 0. By applying Lemma 5.1.3 to the system of linear equations (5.1.38), we finish the proof of Lemma 5.1.5.

277

5.1 Asymptotic expansions for absorption and hitting probabilities

The following lemma is a direct corollary of Lemma 5.1.3 if applied to the hitting .0/ probability vectors 0 f."/ j , j 2 X1 . .k/ hold. Then Lemma 5.1.6. Let conditions A2 , E2 , and P18 ."/

(i) The vector 0 fj ."/ 0 fj

.0/

has the following asymptotic expansion for every j 2 X1 : .0/

D 0 fj

C 0 bj Œ1; 0" C    C 0 bj Œk; 0"k C o."k /;

(5.1.43)

where 0 bj Œr; 0, r D 1; : : : ; k, are finite vectors. (ii) The vector coefficients 0 bj Œr; 0 are given by the recurrence formulas 0 bj Œ0; 0 D .0/ D j UŒ0p.0/ D ŒI  j P.0/ 1 p.0/ 0 fj j j , for r D 0; : : : ; k,   r X b Œr; 0 D UŒ0 e Œr; 0 C EŒq; 0 b Œr  q; 0 : j 0 j j j 0 j

(5.1.44)

qD1

(iii) The vector coefficients 0 bj Œr; 0 can also be calculated for r D 0; : : : ; k by the following formulas: 0 bj Œr; 0 D

r X

j UŒp ej Œr

 p; 0:

(5.1.45)

pD0

Proof. Apply Lemma 5.1.3 to the system of linear equations (5.1.39) for every j 2 is regarded as the vector y."/ . The non-negativity X1.0/ . In this case, the vector p."/ j condition S1 obviously holds. Also, the asymptotic relations (5.1.37) imply that condi.k/ .0/ tion P19 holds for j 2 X1 . Applying Lemma 5.1.3 to the system of linear equations (5.1.39) yields the claims of Lemma 5.1.6.

5.1.5 Pivotal properties of asymptotic expansions for absorption and hitting probabilities We use the following notations for the vectors j b0 Œr; 0 and 0 bj Œr; 0, r D 0; : : : ; k: 2 j b0 Œr; 0

6 D4

j b10 Œr; 0

3

7 :: 5; : j bN 0 Œr; 0

2 0 bj Œr; 0

6 D4

0 b1j Œr; 0

3

7 :: 5: : 0 bNj Œr; 0

Explicit expressions for components of the vectors j b0 Œr; 0, r D 0; : : : ; k, and r D 0; : : : ; k, can be derived from the recurrence formulas (5.1.41) and (5.1.42) or (5.1.44) and (5.1.45), respectively.

0 bj Œr; 0,

278

5

Nonlinearly perturbed semi-Markov processes

Note that if k D 0, Lemmas 5.1.6 and 5.1.5 reduce to Lemma 4.2.1, i.e., it yields convergence of the absorption and hitting probabilities. Thus, j bi0 Œ0; 0 D jfi0.0/ and .0/ .0/ 0 bij Œ0; 0 D 0 fij for i ¤ 0, j 2 X1 . We are interested to clarify pivotal properties of asymptotic expansions for the .0/ cyclic absorption probabilities jfj."/ 0 , j 2 X1 , and the cyclic hitting probabilities ."/ 0 fij ,

i ¤ 0, j 2 X1.0/ . For j 2 X, let ( fi 2 X0 W eij Œn; 0 D 0; n D 0; : : : ; r  1; eij Œr; 0 ¤ 0g Yj;r D fi 2 X0 W eij Œn; 0 D 0; n D 0; : : : ; kg

for r D 1; : : : ; k, for r D k C 1.

are taken to be y."/ . The set Yj;r is regarded as the set Yr if the vectors p.0/ j By the definition, (a) some of the sets Yj;r , r D 0; : : : ; k C 1, could be empty, (b) S Yj;r 0 \ Yj;r 00 D ¿, r 0 ¤ r 00 , but (c) kC1 r D0 Yj;r D X0 . .0/ For j 2 X1 , consider the sets Wj;r D

[

.Yj;r 0 \ Zr 00 /;

r D 0; : : : ; k;

r 0 Cr 00 Dr

Wj;kC1 D

[

.Yj;r 0 \ Zr 00 /:

r 0 Cr 00 kC1 .0/

The case where condition E02 holds is simple. Indeed Z0 D X1 therefore, Z1 ; : : : ; ZkC1 D ¿. Thus, for j 2 X1.0/ , Wj;r D Yj;r ;

r D 0; : : : ; k C 1:

D X0 , and,

(5.1.46)

For integer 0  h  k, introduce two conditions, .h/

O3 : W0;r D ¿, for r  h; and .h/

O4 : W0;r D ¿, for r < h, but W0;h ¤ ¿. The following lemma gives a complete description of pivotal properties of asymp."/ .0/ totic expansions for the cyclic absorption probabilities jfj 0 , j 2 X1 . .k/ hold. Then Lemma 5.1.7. Let conditions A2 , E2 , and P18 .h/

(i) If condition O3 holds for some 0  h  k, then j bj 0 Œn D 0, n D 0; : : : ; h, .0/ j 2 X1 .

5.1 Asymptotic expansions for absorption and hitting probabilities

279

.h/

(ii) If condition O4 holds for some 0  h  k, then j bj 0 Œn D 0, n D 0; : : : ; h  1, .0/ .0/ j 2 X1 , but j bj 0 Œh > 0, j 2 X1 . ."/ Proof. We will again use Lemma 5.1.4 with the vector p."/ 0 replacing the vectors y ."/ and the absorption probabilities jfj."/ 0 replacing the cyclic components xjj . The nonnegativity condition S1 obviously holds. Also, the asymptotic relation (5.1.37) implies .k/ ."/ .h/ .h/ that condition P19 holds for the vector p0 . Conditions O1 and O2 take, respec.h/ .h/ tively, the form of conditions O3 and O4 . Thus, Lemma 5.1.4 just takes the form of Lemma 5.1.7.

The following simple lemma gives a useful information about pivotal properties of the non-cyclic hitting probabilities 0 fij."/ ; i ¤ 0; j 2 X1.0/ . .k/

Lemma 5.1.8. Let conditions A2 , E2 , and P18 hold. Then the asymptotic expansions ."/ .0/ for the hitting probabilities 0 fij , i ¤ 0, j 2 X1 , are all 0-pivotal, i.e., the corre.0/

sponding zero coefficients are positive, 0 fij

.0/

> 0 for all i ¤ 0, j 2 X1 .

Proof. The statement of the lemma immediately follows from the definition of the set .0/ .0/ of recurrent-without-absorption states X1 , according to which we have 0 fij > 0 .0/

for all i ¤ 0, j 2 X1 . It is useful to recall that if conditions A2 , T1 , (a), and E2 are satisfied, then there exists 0 < "1  "0 such that X1.0/ X1."/ for "  "1 . For "  "1 , the hitting and absorption probabilities are connected by the following relation: ."/ 0 fij

."/

D 1  jfi0 ;

i ¤ 0;

.0/

j 2 X1 :

(5.1.47)

Thus, the asymptotic expansion for the vectors of hitting and absorption probabilities satisfy the relation ."/ 0 fj

k k D 0 f.0/ j C 0 bj Œ1; 0" C    C 0 bj Œk; 0" C o." / k k D 1  j f.0/ 0  j b0 Œ1; 0"      j b0 Œk; 0" C o." /;

(5.1.48)

where 1 is a column vector with N components all equal 1. This means that the coefficients in the corresponding asymptotic expansions for .0/ components of these vectors satisfy for i ¤ 0; j 2 X1 the following: ( 1  j bi0 Œ0; 0 for r D 0, (5.1.49) 0 bij Œr; 0 D j bi0 Œr; 0 for r D 1; : : : ; k. This formula gives a much more detailed description of “higher order” pivotal properties of the hitting probabilities.

280

5

Nonlinearly perturbed semi-Markov processes .k/

Lemma 5.1.9. Let conditions A2 , E2 , and P18 hold. Then .0/ (i) If condition O.0/ 3 holds, then 0 bjj Œ0; 0 D 1  j bj 0 Œ0; 0 D 1, j 2 X1 . .h/

.0/

(ii) If condition O3 holds for some 1  h  k, then 0 bjj Œ0; 0 D 1, j 2 X1 .0/ 0 bjj Œn; 0 D 0, 1  n  h, j 2 X1 .

and

.0/ .0/ (iii) If condition O.0/ 4 holds, then 0 bjj Œ0; 0 D 0 fjj D 1 j bj 0 Œ0; 0 > 0, j 2 X1 . .h/

.0/

(iv) If condition O4 holds for some 1  h  k, then 0 bjj Œ0; 0 D 1, j 2 X1 and .0/ 0 bjj Œn; 0 D 0, 1  n  h  1, j 2 X1 but 0 bjj Œh; 0 D j bj 0 Œh; 0 < 0, j 2 X1.0/ . Proof. The lemma is a direct corollary of relation (5.1.49) and Lemmas 5.1.7 and 5.1.8.

5.2 Asymptotic expansions for power moments of hitting times In this section we describe a recursive algorithm used in the asymptotic analysis of power moments of the absorption and hitting times for nonlinearly perturbed semiMarkov processes. The algorithm is important in its own right and makes an essential part of the proof of our main results concerning semi-Markov processes.

5.2.1 Power moment perturbation conditions for transition times Consider the power moments of transition times, for n D 0; 1; : : : ; k, i ¤ 0, j 2 X, Z 1 Z 1 ."/ ."/ ."/ ."/ n ."/ s Qij .ds/ D mij Œnpij ; mij Œn D s n Fij."/ .ds/: pij Œn D 0

0

."/

Note that, by the definition, mij Œ0 D 1, i; j ¤ 0. Note also that the power moments defined above are connected by the following formulas, for n D 0; 1; : : : ; k, i ¤ 0, j 2 X, ."/ ."/ ."/ Œn D pij mij Œn: pij ."/

."/

Also note that mij Œ1 D mij ; i; j ¤ 0, are expectations for the transition times. We further assume that conditions A2 and E2 hold, and also that the following condition holds: .k/

."/

.0/

M13 : pij Œn ! pij Œn < 1 as " ! 0, for n D 0; : : : ; k, i; j ¤ 0.

281

5.2 Asymptotic expansions for power moments of hitting times .k/

The following condition and condition T1 obviously imply condition M13 to hold:

0

.0/ M13.k/ : m."/ ij Œn ! mij Œn < 1 as " ! 0, for n D 0; : : : ; k, i; j ¤ 0. .k/

Condition M13 guarantees that there exists 0 < "2  "1 such that, for i; j ¤ 0, ."/

sup max pij Œk D Mk < 1:

""2 i;j ¤0 ."/

(5.2.1)

."/

By the definition, pij Œ0 D pij , i; j ¤ 0. We assume that the following perturbation condition, which is more general than .k/ P18 , to hold: .k/

."/

.0/

P20 : pij Œn D pij Œn C eij Œ1; n" C    C eij Œk  n; n"kn C o."kn / for n D 0; : : : ; k, i; j ¤ 0, where jeij Œr; nj < 1, r D 1; : : : ; k  n, n D 0; : : : ; k, i; j ¤ 0. .0/

It is convenient to define eij Œ0; n D pij Œn for n D 0; : : : ; k, i; j ¤ 0. .k/

.k/

Note that condition P20 implies condition M13 . ."/ .k/ It is useful to note that moments pi0 Œn are not involved in condition P20 . It is because of these moments do not penetrate formulas for power moments of hitting without absorption times. .k/ In some cases, it can be more convenient to use the perturbation condition P18 for ."/ the transition probabilities pij , i; j ¤ 0, and, additionally, the following perturbation ."/

condition imposed on the moments of transition times, mij Œn, n D 1; : : : ; k, i; j ¤ 0: .k/

."/

.0/

P21 : mij Œn D mij Œn C vij Œ1; n" C    C vij Œk  n; n"kn C o."kn / for n D .0/

1; : : : ; k, i; j ¤ 0 where (a) mij Œn < 1, n D 1; : : : ; k, i; j ¤ 0, and (b) jvij Œr; nj < 1, r D 1; : : : ; k  n, n D 1; : : : ; k, i; j ¤ 0. .0/

Define vij Œ0; n D mij Œn for i; j ¤ 0, n D 1; : : : ; k. .0/

For k D 0, condition P21 should be interpreted as an “empty” condition that holds automatically. .k/ .k/ .k/ .k/ If conditions P18 and P21 are satisfied, then conditions M13 and P21 also hold. Moreover, in such a case, the corresponding asymptotic expansions are connected with .k/ the asymptotic expansion in condition P20 by the following formulas for i; j ¤ 0: 8 < eij Œr; 0 r P eij Œr; n D : eij Œm; 0vij Œr  m; n mD0

.k/

if r D 0; : : : ; k, n D 0, if r D 0; : : : ; k  n, n D 1; : : : ; k:

We will rewrite condition P20 in a matrix form.

(5.2.2)

282

5

Nonlinearly perturbed semi-Markov processes

For n D 0; 1; : : : ; k, introduce the matrices 3 ."/ ."/ p11 Œn : : : p1N Œn 7 6 :: :: :: P."/ Œn D 4 5; : : : ."/ ."/ pN1 Œn : : : pN N Œn 2

and, for r D 0; : : : ; k  n, n D 1; : : : ; k, 2 e11 Œr; n 6 :: EŒr; n D 4 :

3 e1N Œr; n 7 :: 5: : eN1 Œr; n : : : eN N Œr; n ::: :: :

.k/

Condition P20 can be rewritten in the following equivalent matrix form: .k/ : P."/ Œn D P.0/ Œn C EŒ1; n" C    C EŒk  n; n"k C o."kn /, where P.0/ Œn P20 and EŒr; n, r D 1; : : : ; k  n, n D 1; : : : ; k are finite matrices.

For n D 0; : : : ; k, j ¤ 0, consider the matrices 2 ."/ ."/ ."/ ."/ p11 Œn : : : p1.j 1/ Œn 0 p1.j C1/ Œn : : : p1N Œn 6 :: :: :: :: :: ."/ 6 j P Œn D 4 : : : : : ."/ ."/ ."/ ."/ pN1 Œn : : : pN.j 1/ Œn 0 pN.j C1/ Œn : : : pN N Œn and also, for r D 0; : : : ; k  n, n D 1; : : : ; k, and j ¤ 0, 2 e11 Œr; n : : : e1.j 1/ Œr; n 0 e1.j C1/ Œr; n 6 :: :: :: :: j EŒr; n D 4 : : : :

3 7 7; 5

3 e1N Œr; n 7 :: 5: : eN1 Œr; n : : : eN.j 1/ Œr; n 0 eN.j C1/ Œr; n : : : eN N Œr; n :::

.k/

If j ¤ 0, condition P20 gives the following matrix asymptotic expansions for n D 0; : : : ; k: jP

."/

Œn D j P.0/ Œn C j EŒ1; n" C    C j EŒk  n; n"kn C o."kn /: .0/

(5.2.3)

It is convenient to define eij Œ0; n D pij Œn, n D 0; : : : ; k, i; j ¤ 0, and the matrices EŒ0; n D P.0/ Œn, j EŒ0; n D j P.0/ Œn, n D 0; : : : ; k, j ¤ 0. For r D k  n, n D 0; : : : ; k, and j ¤ 0, consider the vectors 3 2 ."/ 3 2 p1j Œn e1j Œr; n 7 6 7 6 ."/ :: :: 7; pj Œn D 6 ej Œr; n D 4 5: : : 5 4 ."/ eNj Œr; n p Œn Nj

283

5.2 Asymptotic expansions for power moments of hitting times .k/

Condition P20 gives the following vector asymptotic expansion for n D 0; : : : ; k and j ¤ 0: .0/ kn C o."kn /: p."/ j Œn D pj Œn C ej Œ1; n" C    ej Œk  n; n"

(5.2.4)

.0/

Recall that eij Œ0; n D pij Œn, n D 0; : : : ; k, i; j ¤ 0. In the sequel, it will be .0/

convenient to define the vectors ej Œ0; n D pj Œn, n D 0; : : : ; k, j ¤ 0.

5.2.2 Recursive systems of linear equations for power moments of hitting times Let us also introduce power moments for the hitting times, ."/ 0 Mij Œn

D Ei .j."/ /n .j."/  0."/ /;

n D 0; 1; : : : ; k;

i; j ¤ 0:

By the definition we have ."/ 0 Mij Œ0

D Pi fj."/  0."/ g D 0 fij."/ ;

i; j ¤ 0:

Recall again that if conditions A2 , T1 , (a), and E2 hold, then there exists "1 > 0 such that X1.0/ X1."/ for "  "1 . Therefore, (a) the hitting probabilities satisfy ."/ .0/ .k/ 0 fij > 0, i ¤ 0, j 2 X1 . Note also that condition M13 implies that (c) there exists ."/

0 < "2  "1 such that pij Œk  Mk < 1 for "  "2 and i; j ¤ 0. As was shown in Subsection 4.2.3, (c) there exist 0 < L < 1 and 0 < "3  "2 such ."/ ."/ that maxl¤0 Pl fj ^ 0 > ng  LŒn=m for n  1 and "  "3 . These relations imply that for "  "3 ."/

sup 0 Mij Œk < 1;

""3

.0/

i ¤ 0;

j 2 X1 :

(5.2.5)

Indeed, using relation (4.2.17), we get for "  "3 , ."/ 0 Mij Œk





1 X

n

X

k

nD1

i0 Di;i1 ;:::;in1 ¤0;j; in Dj

1 X

n

."/

nk Mk max Pl j ^ 0 > l¤0

nD1

 Mk

."/

! n n Y ."/ 1 X ."/ mir1 ir Œk pir1 ir n

1 X

nk LŒ

n1 2 1=m

rD1

hn  1 2

rD1

io 1

< 1:

(5.2.6)

nD1 ."/

The moments Mij Œn, i ¤ 0, satisfy the following system of linear equations for .0/

every "  "3 , j 2 X1 , n D 0; 1; : : : ; k: X ."/ ."/ ."/ ."/ pil 0 Mlj Œn; 0 Mij Œn D qij Œn C l¤0;j

i ¤ 0;

(5.2.7)

284

5

Nonlinearly perturbed semi-Markov processes

where ."/ qij Œn

D

n X

."/ pij Œn C

mD1

and Cnm D

X

Cnm

."/

."/

pil Œm 0 Mlj Œn  m;

i ¤ 0;

(5.2.8)

l¤0;j

nŠ ; mŠ.n  m/Š

0  m  n < 1;

0Š D 1: .0/

These systems can be rewritten in a matrix form. To do this, introduce for j 2 X1 and n D 0; : : : ; k, the vectors 2 2 ."/ 3 3 ."/ q1j Œn 0 M1j Œn 6 6 7 7 ."/ :: :: 6 6 7; 7: q."/ 0 mj Œn D 4 : : j Œn D 4 5 5 ."/ ."/ q M Œn Œn 0 Nj Nj

Then, the system of linear equations (5.2.7) can be rewritten in the following form .0/ for every j 2 X1 and n D 0; 1; : : : ; k: ."/ 0 mj Œn

."/ ."/ D q."/ 0 mj Œn: j Œn C j P

(5.2.9)

These are systems of hitting type considered in Subsection 5.1.3, with the same coefficient matrix j P."/ . As was pointed out in this subsection, conditions A2 , E2 , T1 (a), and S1 imply that there exists "0 > 0 such that for every "  "0 any hitting type system of linear equation with the coefficient matrix j P."/ has a unique solution. .k/ should be added as to provide finiteness of the k-order moments Condition M13 .0/ of the transition times. Obviously, the vectors q."/ j Œn, n D 0; : : : ; k, j 2 X1 , have non-negative components. .0/ A solution of system (5.2.9) has the following form for "  "3 , j 2 X1 , and n D 0; : : : ; k: ."/ 0 mj Œn

."/

D ŒI  j P."/ 1 qj Œn:

(5.2.10)

As was mentioned above, systems (5.2.9) have the same coefficient matrix j P."/ .0/ ."/ for a given j 2 X1 but different inhomogeneous terms qj Œn for n D 0; 1; : : : ; k. These systems should be solved recursively. First, system (5.2.7) should be solved for n D 0. Note that it is actually a system of ."/ ."/ linear equations for the hitting probabilities 0 fij D 0 Mij , i ¤ 0. Second, system (5.2.7) should be solved for n D 1. Note that the expressions for P ."/ ."/ ."/ ."/ the inhomogeneous terms qij Œ1 D pij Œ1C l¤0;j pil Œ1Mlj Œ0, i ¤ 0, in (5.2.8) ."/

include the solutions Mij Œ0, i ¤ 0, of systems (5.2.7) for n D 0.

285

5.2 Asymptotic expansions for power moments of hitting times

This recursive procedure should be repeated for n D 1; : : : ; k. The expressions ."/ Œn given in (5.2.8) include the solutions Mij."/ Œm, for the inhomogeneous terms qij i ¤ 0, of systems (5.2.7) for m D 0; 1; : : : ; n  1. .0/ Formula (5.2.8) can be rewritten in the following vector form for every j 2 X1 and n D 0; 1; : : : ; k: ."/

."/

qj Œn D pj Œn C

n X

."/

Cnm j P."/ Œm0 mj Œn  m:

(5.2.11)

mD1

Using formula (5.2.11) we can rewrite formula (5.2.10) in a form that uses the recurrence procedure for calculating solutions of the system of linear equations (5.2.7) .0/ for every j 2 X1 and n D 0; 1; : : : ; k, ."/ 0 mj Œn

."/

D ŒI  j P."/ 1 qj Œn ."/

D ŒI  j P."/ 1 pj Œn C

n X

."/

Cnm ŒI  j P."/ 1 j P."/ Œm0 mj Œn  m:

(5.2.12)

mD1

5.2.3 Asymptotic expansions for power moments of hitting times The following lemma is a corollary of Lemma 5.1.3 applied to the vectors of power ."/ .0/ moments of the hitting times, 0 mj Œn, for j 2 X1 and n D 0; 1; : : : ; k. .k/ .k/ , and P20 , hold. Then Lemma 5.2.1. Let conditions A2 , E2 , M13 .0/ (i) The vector 0 m."/ j Œn has the following asymptotic expansion for every j 2 X1 and n D 0; : : : ; k: ."/ 0 mj Œn

.0/

D 0 mj Œn C 0 bj Œ1; n" C    C 0 bj Œk  n; n"k C o."kn /; (5.2.13)

where 0 bj Œr; n, r D 1; : : : ; k  n, n D 0; : : : ; k, are finite vectors. (ii) The vector coefficients 0 bj Œr; n are given by the recurrence formulas 0 bj Œ0; 0 D .0/ .0/ .0/ 1 .0/ 0 mj Œ0 D j UŒ0qj Œ0 D ŒI j P  pj and for r D 0; : : : ; k n (for given n) and sequentially for every n D 0; : : : ; k, 0 bj Œr; n

D j UŒ0.ejŒr; n C

n X mD1

C

r X pD1

Cnm

r X

j EŒq; m 0 bj Œr

 q; n  m

qD0

Ej Œp; 0 0 bj Œr  p; n/:

(5.2.14)

286

5

Nonlinearly perturbed semi-Markov processes

(iii) The vector coefficients 0 bj Œr; n also can be calculated by the recurrence formulas for r D 0; : : : ; k  n (for given n) and sequentially for every n D 0; : : : ; k, 0 bj Œr; n D

r X

j UŒqej Œr

qD0

C

n X

Cnm

mD1

0

r X

@

qD0

 q; n

q X

1 j UŒpj EŒq

 p; mA 0 bj Œr  q; n  m: (5.2.15)

pD0

.0/

Proof. Fix some j 2 X1 . The asymptotic expansions (5.2.13) are constructed recur."/ sively for the vectors 0 mj Œn where n D 0; : : : ; k. ."/

If n D 0, then 0 mj Œ0 D .k/ P20

."/ 0 fj .

The asymptotic expansion given for n D 0

.k/ coincides with the asymptotic expansion given in condition P18 . in condition ."/ The asymptotic expansion (5.2.13) for the vectors 0 mj Œ0 and formulas (5.2.14) and (5.2.15) for the corresponding vector coefficients just coincide, respectively, with the ."/ asymptotic expansion (5.1.43) for the vectors 0 fj and formulas (5.1.44) and (5.1.45) for the corresponding vector coefficients. Note that this is an .N; 1/-matrix .0; k/expansion. ."/ Now we can write an asymptotic expansion for the vectors q."/ j Œ1 D pj Œ1 C jP

."/

."/

Œ1  0 mj Œ0 by using Lemma 8.1.2. This will be an .N; 1/-matrix .0; k  1/."/

expansion. Then Lemma 5.1.3 can be applied to the vectors 0 mj Œ1 D ŒI  j P."/ 1 

q."/ j Œ1. The resulted asymptotic expansion is an .N; 1/-matrix .0; k  1/-expansion. The recursive procedure can be repeated for n D 1; : : : ; k. At the step n, the ."/ corresponding expansion for the vectors qj Œn given by formula (5.2.11) make an .N; 1/-matrix .0; k  n/-expansion that takes the following form: kn q."/ C o."kn /; j Œn D qj Œ0; n C qj Œ1; n" C    C qj Œk  n; n"

(5.2.16)

where the coefficients qj Œr; n for r D 0; : : : ; k  n are given by the formulas qj Œr; n D ej Œr; n C

n X mD1

Cnm

r X

j EŒq; m0 bj Œr

 q; n  m:

(5.2.17)

qD0

."/ 1 ."/ Then, we apply Lemma 5.1.3 to the vectors 0 m."/ j Œn D ŒI  j P  qj Œn. The obtained asymptotic expansion makes an .N; 1/-matrix .0; k  n/-expansion given by the asymptotic expansion (5.2.13) and formulas (5.2.14) and (5.2.15) for the corresponding vector coefficients.

Remark 5.2.1. It should be noted that both formulas (5.2.14) and (5.2.15) for the corresponding vector coefficients in the asymptotic expansions (5.2.13) have a recursive character. However, they differ in their structure. Formulas (5.2.14) are used for

287

5.2 Asymptotic expansions for power moments of hitting times

calculating the coefficients 0 bj Œr; n and 0 bj Œl; m, l D 0; : : : ; r, m D 0; : : : ; n  1. While formulas (5.2.15) give the coefficients 0 bj Œr; n and 0 bj Œl; m, l D 0; : : : ; r, m D 0; : : : ; n  1, and also the coefficients 0 bj Œl; n, l D 0; : : : ; r  1. Q define the Introducing the integer parameters nQ  1 and kQn  0, n D 0; : : : ; n, vector parameter kQ D .n, Q kQ0 ; : : : ; kQnQ /. Let us now assume that the following perturbation condition, which is more general .k/ than P20 , is satisfied: Q Q Q ."/ .0/ P.22k/ : pij Œn D pij Œn C eij Œ1; n" C    C eij ŒkQn ; n"kn C o."kn / for n D 0; : : : ; n, Q

.0/ Œn < 1, n D 0; : : : ; n, Q i; j ¤ 0 and jeij Œr; nj < 1, i; j ¤ 0, where pij r D 1; : : : ; kQn , n D 0; : : : ; n, Q i; j ¤ 0.

.kQ / .k/ Condition P22 reduces to condition P20 if nQ D k; kQn D k  n, n D 0; : : : ; k. The following lemma generalises Lemma 5.2.1 and reduces to this lemma in the case mentioned above. .kQ /

.Qn/

Lemma 5.2.2. Let conditions A2 , E2 , M13 , and P22 hold. Then .0/ (i) The vector 0 m."/ Q has the following j Œn, for every j 2 X1 and n D 0; : : : ; n, asymptotic expansion: ."/ 0 mj Œn

.0/

D 0 mj Œn C 0 bj Œ1; n" C    C 0 bj Œkn ; n"kn C o."kn /; (5.2.18)

where kn D min.kQ0 ; : : : ; kQn /, n D 0; : : : ; n, Q and 0 bj Œr; n, r D 1; : : : ; kn , n D 0; : : : ; n, Q are finite vectors. (ii) The vector coefficients 0 bj Œr; n are given by the recurrence formulas 0 bj Œ0; 0 D .0/ .0/ .0/ 1 .0/ and for r D 0; : : : ; kn (for given 0 mj Œ0 D j UŒ0qj Œ0 D ŒI  j P  pj n) and sequentially for every n D 0; : : : ; n, Q 0 bj Œr; n

n X

D j UŒ0.ejŒr; n C

Cnm

mD1 r X

C

r X

j EŒq; m0 bj Œr

 q; n  m

qD0

Ej Œp; 00 bj Œr  p; n/:

(5.2.19)

pD1

(iii) The vector coefficients 0 bj Œr; n can also be calculated by the recurrence formulas for r D 0; : : : ; kn (for given n) and sequentially for every n D 0; : : : ; n, Q 0 bj Œr; n

D

r X

j UŒqej Œr

qD0

C

n X mD1

Cnm

r X qD0

0 @

 q; n

q X

pD0

1 j UŒpj EŒq

 p; mA 0 bj Œr  q; n  m: (5.2.20)

288

5

Nonlinearly perturbed semi-Markov processes

Proof. The proof repeats the proof of Lemma 5.2.1. Lemma 5.2.2 differs from Lemma 5.2.1 in the orders of the corresponding asymptotic expansions for the power moments of hitting times, 0 m."/ j Œn. These orders also should be calculated recursively.

."/ if n D 0. The asymptotic expansion As was mentioned above, 0 m."/ j Œ0 D 0 fj .kQ /

given for n D 0 in condition P22 coincides with the asymptotic expansion given in .k/ condition P18 if we choose k D kQ0 . The asymptotic expansion (5.2.18) for the vec."/ ."/ tors 0 mj Œ0 D ŒI  j P."/ 1 qj Œ0 and formulas (5.2.19) and (5.2.20) for the corresponding vector coefficients just coincide, respectively, with the asymptotic expansion ."/ (5.1.43) for the vectors 0 fj and formulas (5.1.44) and (5.1.45) for the corresponding vector coefficients. Note that this is an .N; 1/-matrix .0; k0 /-expansion, where k0 D kQ0 . Indeed, the order of this expansion is the minimum of orders of the asymp."/ totic expansions for the matrices ŒI  j P."/ 1 and the vectors qj Œ0. Both these .kQ / expansions have, according to condition P , the order kQ0 . 22

."/ Now we can write an asymptotic expansion for the vectors q."/ j Œ1 D pj Œ1 C ."/

."/

Œ1  0 mj Œ0 by using Lemma 8.1.2. This will be an .N; 1/-matrix .0; k1 /-expansion, where k1 D min.kQ0 ; kQ1 /. Indeed, according to Lemma 5.1.3, the order of this expansion is the minimum of the orders of the asymptotic expansions for the vectors ."/ ."/ pj Œ1, the matrices j P."/ Œ1, and the vectors 0 mj Œ0 that are, respectively, kQ1 , kQ1 , and kQ0 . This minimum is k1 D min.kQ0 ; kQ1 /. Then, Lemma 5.1.3 can be applied to jP

."/

."/

the vectors 0 mj Œ1 D ŒI  j P."/ 1 qj Œ1. The resulted asymptotic expansion is an .N; 1/-matrix .0; k1 /-expansion given in formula (5.2.18) for n D 1. According to Lemma 5.1.3, the order of this expansion is the minimum of orders of the asymptotic expansions for the matrices ŒI  j P."/ 1 and the vectors q."/ j Œ1. These expansions Q Q Q have, respectively, the order k1 , where k1 D min.k0 ; k1 /. This minimum is k1 . The recurrence procedure can be repeated for n D 1; : : : ; n, Q which will yield the corresponding asymptotic expansions given in (5.2.18).

5.2.4 Pivotal properties of asymptotic expansions for power moments of hitting times We will use the following notations for the vectors 0; : : : ; n: Q 2 0 b1j Œr; n 6 :: 0 bj Œr; n D 4 : 0 bNj Œr; n

0 bj Œr; n,

r D 0; : : : ; kn , n D

3 7 5:

Explicit expressions for components of the vectors 0 bj Œr; n, r D 0; : : : ; kn , n D 0; : : : ; n, Q can be derived from the recurrence formulas (5.2.14) and (5.2.15), respectively.

5.3 Asymptotic expansions for power-exponential moments of hitting times

289

."/

Q We are interested in finding the cyclic power moments 0 Mjj Œn, n D 0; : : : ; n, .0/

j 2 X1 . We will limit our consideration to the case where condition I4 is satisfied, that is, .0/ .0/ there exists a state i 2 X1 such that Fi .0/ < 1. This condition guarantees that the corresponding semi-Markov process ."/ .t /, t  0, has a finite number of transitions in every finite interval for " small enough. Q

n/ .k/ Lemma 5.2.3. Let conditions A2 , E2 , I4 , M.Q 13 , and P22 hold. Then the asymptotic expansions for the power moments of the hitting probabilities 0 Mjj."/ Œn, n D 0; : : : ; n, Q

j 2 X1.0/ are all 0-pivotal, i.e., the corresponding zero coefficients satisfy 0 Mjj.0/ Œn D 0 bjj Œ0; n

> 0 for all n D 0; : : : ; nQ and j 2 X1.0/ .

Proof. Condition I4 implies that (c) there exists a state i 0 2 X1.0/ , and j 0 ¤ 0 such .0/ .0/ .0/ that (d) pi 0 j 0 > 0 and (e) 1  Fi 0 j 0 .0/ > 0. Let us choose an arbitrary state j 2 X1 . 0 There are two cases to be considered. The first case is where i D j . Here, (d) .0/ .0/ and (e) imply that (f) 1  0 Gjj .0/ > pi 0 j 0 .1  Fi 0 j 0 .0// > 0. The second case is

where i 0 ¤ j . In this case, (g) the cyclic hitting probability satisfies 0jfj.0/ i 0 > 0, since i 0 is a recurrent-without-absorption state. Using (d), (e), and (g) we get that (h) .0/ .0/ .0/ 1  0 Gjj .0/ > 0jfj i 0 pi 0 j 0 .1  Fi 0 j 0 .0// > 0. Clearly, (f) as well as (h) implies that .0/

.0/

.0/

Q The relation 0 Mjj Œ0 > 0, j 2 X1 , was proved in (i) 0 Mjj Œn > 0, n D 1; : : : ; n. Lemma 5.1.8.

5.3 Asymptotic expansions for power-exponential moments of hitting times In this section, we describe a recurrence algorithm for a use in the asymptotic analysis of mixed power-exponential moments of hitting times for nonlinearly perturbed semiMarkov processes. The algorithm is also important by itself and is an essential in the proof of our main results concerning semi-Markov processes.

5.3.1 Asymptotic expansions for -potential matrices Let us recall some notations introduced in Section 4.4 for moment generating functions, Z 1 Z 1 ."/ ."/ ."/ ."/ pij ./ D e s Qij .ds/; ./ D e s Fij .ds/;  2 R1 ; i; j ¤ 0: ij 0

0

The moment generating functions introduced above are connected by the following formulas, for  2 R1 and i; j ¤ 0, ."/

."/

pij ./ D pij

."/ ij ./:

290

5

Nonlinearly perturbed semi-Markov processes

We assume that the following condition holds: ."/

C17 : There exists ı > 0 such that lim0"!0 pij .ı/ < 1, i; j ¤ 0. Condition C17 is a reduced form of condition C12 where the corresponding asymptotic relation is also required for i ¤ 0, j D 0. According to Remark 4.5.3, these additional asymptotic relations are not required when considering moment generating functions of hitting times without absorption. The following condition obviously implies condition C17 be satisfied: C017 : There exists ı > 0 such that lim0"!0

."/ ij .ı/

< 1, i; j ¤ 0.

Condition C17 guarantees that for every 0  ˇ < ı there exists "00 D "00 .ˇ/ > 0 such that, for   ˇ, ."/

sup max pij ./ < 1:

""00

(5.3.1)

i;j ¤0

Let us also assume the following condition (introduced in Subsection 4.4.4): T2 :

."/

.0/

(a) pij ! pij

as " ! 0, i; j ¤ 0;

."/ .0/ (b) Qij ./ ) Qij ./ as " ! 0, i; j ¤ 0.

The following condition obviously implies condition T2 be satisfied: T02 :

."/

.0/

(a) pij ! pij ."/

as " ! 0; i; j ¤ 0; .0/

(b) Fij ./ ) Fij ./ as " ! 0, i; j ¤ 0. Condition T2 is a reduced form of condition T1 , where the corresponding asymptotic relation is also required for i ¤ 0; j D 0. These asymptotic relations are not required when considering moment generating functions of hitting times without absorption. Conditions T2 and C17 imply that, for i; j ¤ 0 and   ˇ, ."/ .0/ ./ ! pij ./ as " ! 0: pij

(5.3.2)

We assume that the following perturbation condition is satisfied for some   ˇ: .;k/ P23 :

."/

.0/

pij ./ D pij ./ C eij Œ; 1; 0" C    C eij Œ; k; 0"k C o."k / for i; j ¤ 0, where jeij Œ; r; 0j < 1, r D 1; : : : ; k, i; j ¤ 0. .0/

It is convenient to define eij Œ; 0; 0 D pij Œ; 0 for i; j ¤ 0,   ˇ. ."/

It is useful to note that moment generating functions pi0 ./ are not involved in .;k/ condition P23 . It is because of they do not penetrate formulas for power moment generating functions for hitting without absorption times. .k/ In some cases, it can be more convenient to use the perturbation condition P18 for ."/ the transition probabilities pij , i; j ¤ 0, and, additionally, the following perturbation

5.3 Asymptotic expansions for power-exponential moments of hitting times

condition imposed on moments of the transition times .;k/ P24 :

."/ ij ./,

291

i; j ¤ 0:

."/ ij ./

D ij.0/ ./ C vij Œ; 1; 0" C    C vij Œ; k; 0"k C o."k / for i; j ¤ 0, where jvij Œ; r; 0j < 1, r D 1; : : : ; k, i; j ¤ 0.

Define vij Œ; 0; 0 D

.0/ ij ./

for i; j ¤ 0,   ˇ. .k/

.;k/

.;k/

It can be proved that conditions P18 and P24 imply that condition P23 holds. .k/ Moreover, the corresponding asymptotic expansions, assuming that conditions P18 .;k/ and P24 hold, are connected for i; j ¤ 0 with the asymptotic expansion in condition .;k/ P23 by the following formulas: eij Œ; r; 0 D

r X

eij Œr; 0vij Œ; r  m; 0;

r D 0; : : : ; k:

(5.3.3)

mD0 .;k/

We will rewrite condition P23 in a matrix form. For r D 0; 1; : : : ; k, consider the matrices 3 2 ."/ ."/ p11 ./ : : : p1N ./ 7 6 :: :: :: P."/ ./ D 4 5; : : : ."/

."/

pN1 ./ : : : pN N ./ and

3 e11 Œ; r; 0 : : : e1N Œ; r; 0 7 6 :: :: :: EŒ; r; 0 D 4 5: : : : eN1 Œ; r; 0 : : : eN N Œ; r; 0 2

.;k/

Condition P23 .;k/ P23 :

can be rewritten in the following equivalent matrix form:

P."/ ./

D P.0/ ./ C EŒ; 1; 0" C    C EŒ; k; 0"k C o."k /, where EŒ; r; 0, r D 0; : : : ; k are finite matrices. For j ¤ 0, set 2 ."/ 3 ."/ ."/ ."/ p11 ./ : : : p1.j 1/ ./ 0 p1.j C1/ ./ : : : p1N ./ 6 7 :: :: :: :: :: ."/ 6 7; j P ./ D 4 : : : : : 5 ."/ ."/ ."/ ."/ pN1 ./ : : : pN.j 1/ ./ 0 pN.j C1/ ./ : : : pN N ./

and also, for r D 1; : : : ; k and j ¤ 0, j EŒ; r; 0

3 e11 Œ; r; 0 : : : e1.j 1/ Œ; r; 0 0 e1.j C1/ Œ; r; n : : : e1N Œ; r; 0 7 6 :: :: :: :: :: D4 5: : : : : : eN1 Œ; r; 0 : : : eN.j 1/ Œ; r; 0 0 eN.j C1/ Œ; r; 0 : : : eN N Œ; r; 0 2

292

5 .;k/

Condition P23 jP

."/

Nonlinearly perturbed semi-Markov processes

, for j ¤ 0, gives the following matrix asymptotic expansions:

Œ; 0 D j P.0/ Œ; 0 C j EŒ; 1; 0" C    C j EŒ; k; 0"k C o."k /:

(5.3.4)

.0/ It is convenient to define eij Œ; 0; 0 D pij ./, i; j ¤ 0, and the matrices .0/ .0/ EŒ; 0; 0 D P ./ and j EŒ; 0; 0 D j P ./, j ¤ 0. We also assume that the following condition holds:

C18 : There exist i 2 X1.0/ and ˇi 2 .0; ı, for ı defined in condition C17 , such that: .0/

(a): 0 i i .ˇi / 2 .1; 1/; .0/ .ˇi / < 1, k 2 X2.0/ . (b): 0 ki

Condition C18 is a variant of condition C13 in which ı defined in condition C12 is replaced with ı defined in condition C17 . It should be mentioned that necessary and sufficient conditions for condition C18 to hold, formulated in terms of so-called test functions, are given in Lemma 4.5.5. In what follows we assume that conditions A2 , T2 , E2 , C17 , and C18 hold. As was shown in Lemmas 4.5.6 and 8.1.4, these conditions imply that (a) there exist 0 < ˇ < ˇi , where ˇi is from condition C18 , and 0 < "01 D "1 .ˇ/  "00 .ˇ/ such that, .0/ for j 2 X1 , sup j .j P."/ .ˇ//n j ! 0 as n ! 1:

""01

(5.3.5)

Relation (5.3.5) implies that (b) there exists the inverse matrix ŒI  j P."/ ./1 for .0/ all   ˇ, j 2 X1 , and "  "01 . Introduce a notation for this matrix, jU

."/

./ D kj u."/ ./k D ŒI  j P."/ ./1 : il

Elements of the matrix j U."/ ./ have a clear probability meaning, namely, for i; l ¤ .0/ 0, j 2 X1 , ."/ j uil ./

D Ei ıj."/ ./; l

where ."/

."/ ıj l ./

."/

j ^0

D

X

e 

."/ .n1/

."/

ı.n1 ; l/

nD1

D

1 X nD1

e 

."/ .n1/

.j."/ ^ 0."/ > n  1; ."/ n1 D l/:

(5.3.6)

5.3 Asymptotic expansions for power-exponential moments of hitting times

293

We refer to the matrix j U."/ ./ as a -potential matrix associated with the matrix P."/ ./. It is useful to note that j U."/ .0/ D j U."/ , i.e., 0-potential matrices coincide with the corresponding usual potential matrices studied in Section 5.1. The following lemma, which is a direct corollary of Lemma 8.1.2, gives an asymptotic matrix expansion for the potential matrices j U."/ ./, j 2 X1.0/ . In Lemmas 5.3.1–5.3.4, ˇ is defined in (5.3.5). .;k/

Lemma 5.3.1. Let conditions A2 , T2 , E2 , C17 , and C18 hold, and also condition P23 be satisfied for some   ˇ. Then .0/

(i) The -potential matrices j U."/ ./, for every j 2 X1 , have the following asymptotic matrix expansions: jU

."/

./ D j U.0/ ./ C j UŒ; 1" C    C j UŒ; k"k C o."k /;

(5.3.7)

where j UŒ; r, r D 1; : : : ; k, are finite matrices. (ii) The matrix coefficients j UŒ; r are given by the following recurrence formulas .0/ .0/ 1 and, for r D 1; : : : ; k, j UŒ; 0 D j U ./ D ŒI  j P ./ j UŒ; r

D j UŒ; 0

r X

j EŒ; q; 0j UŒ; r

 q:

(5.3.8)

qD1

Proof. Relations (5.3.4) and (5.3.5) imply that conditions of Lemma 8.1.2 are satisfied .0/ by the nonlinearly perturbed matrices I  j P."/ ./ for every j 2 X1 . By applying statement (iv) of Lemma 8.1.2 to these matrices, we get the claims of Lemma 5.3.1.

5.3.2 Pivotal properties of asymptotic expansions for -potential matrices Let us use the following notations for the matrices j UŒ; r, r D 0; : : : ; k: 3 2 j u11 Œ; r : : : j u1N Œ; r 7 6 :: :: :: 5: j UŒ; r D 4 : : : j uN1 Œ; r : : : j uN N Œ; r Explicit expressions for elements of the matrices j UŒ; r, r D 0; : : : ; k, can be easily derived from recurrence formulas (5.3.8). We would like to clarify pivotal properties of asymptotic expansions for the com."/ ponents j uil ./ of elements of the potential matrices j U."/ ./. A description of such properties is especially important for cyclic elements of these matrices; these are ele./ for the recurrent-without-absorption states j 2 X1.0/ . ments j uj."/ l

294

5

Nonlinearly perturbed semi-Markov processes

We are going to show that, under a natural assumption that conditions of Lemma .k/ hold, j uj."/ ./ and j uj."/ have the same pivotal 5.3.1 and the perturbation condition P18 l l properties. For a pair of states .i; j /, where i; j ¤ 0, define a parameter, called a local asymp."/ totic -order, via coefficients in the asymptotic expansion for the quantities pij ./ .;k/

given in condition P23 , ( r if eij Œ; n; 0 D 0; n < r but eij Œ; r; 0 > 0, for r D 0; : : : ; k, r .i; j / D k C 1 if eij Œ; n; 0 D 0; n  k. Consider an .i; j /-chain .i0 ; i1 ; : : : ; im / and define a parameter, called a chain asymptotic -order, via the corresponding local asymptotic -order of sequential pairs of states in this chain, ( P r .in1 ; in / D r for r D 0; : : : ; k, r if m r .i0 ; i1 ; : : : ; im / D PnD1 m k C 1 if nD1 r .in1 ; in /  k C 1. Finally, we define a global asymptotic -order for a pair of states .i; j /, where i; j ¤ 0, via the chain asymptotic -orders of the .i; j /-chains, rO .i; j / D

min

0  .i00 ;1 ;:::;im 0 /2Rij

0 r .i00 ; : : : ; im 0 /:

Consider the sets Zj;r; D ( fl 2 X0 W j uj l Œ; n D 0; n D 0; : : : ; r  1; j uj l Œr ¤ 0g fl 2 X0 W j uj l Œ; n D 0; n D 0; : : : ; kg

for r D 1; : : : ; k, for r D k C 1.

.k/

.;k/

Lemma 5.3.2. Let conditions A2 , T2 , E2 , C17 , C18 , P18 hold, and condition P23 satisfied for some   ˇ. Then

be

(i) r .i; j / D r.i; j /, i; j ¤ 0, and r .i0 ; i1 ; : : : ; im / D r.i0 ; i1 ; : : : ; im /,  , i; j ¤ 0. .i0 ; i1 ; : : : ; im / 2 Rij .0/

(ii) rO .i; j / D r.j; O l/ D rO .l/ does not depend on the choice of the state j 2 X1 for every state l ¤ 0. (iii) Zj;r; D Zr , j ¤ 0, r D 0; : : : ; k C 1. .0/

Proof. Take an arbitrary i; j ¤ 0. By the definition, (a) 0 < pij ./ < 1 and, ."/

.0/

by relation (5.3.2), (b) pij ./ ! pij ./ as " ! 0. It follows from statements (a) .k/

.;k/

and (b) and conditions P18 , P23

that either (c) there exists 0  r  k such that

295

5.3 Asymptotic expansions for power-exponential moments of hitting times ."/

."/

lim"!0 "r pij D eij Œr; 0 > 0 and lim"!0 "r pij ./ D eij Œ; r; 0 > 0, i.e., both ."/

."/

expansions given in these conditions are r-pivotal, or (d) pij D o."k / and pij ./ D o."k /. Thus, claim .˛/ is verified. Statements .ˇ/ and ./ are obvious corollaries of .˛/. Both j P."/ ./ and j P."/ are matrices with non-negative elements and with zero element in the column j . The matrices j P."/ ./ can be considered as usual potential ."/ ."/ matrices j P."/ but with elements pij ./ instead of elements pij . Note also that the

."/ ."/ ./ and pij take positive or zero values simultaneously. The asympelements pij totic -orders r .i; j /; r .i0 ; i1 ; : : : ; im / and rO .i; j / replace, respectively, the usual asymptotic orders r.i; j /; r.i0 ; i1 ; : : : ; im /, and r.i; O j /. .;k/ Conditions A2 , T2 , E2 , C17 , C18 , P23 imply that the conditions of Lemma 5.1.2 are satisfied for these matrices. Therefore, claims (i)–(iv) of Lemma 5.1.2 are verified for the matrices j P."/ ./.

In particular, (e) the global asymptotic orders rO .i; j / D rO .j /, i 2 X1.0/ , j ¤ 0, do .0/ not depend on the choice of i 2 X1 , and (f) the sets Zj;r; D Zr; , r D 0; : : : ; k C 1, .0/ do not depend on the choice of j 2 X1 . By the definition, Zr; D fj ¤ 0 W rO .j / D O /, j ¤ 0. Thus, Zr; D Zr , rg. But statement ./ implies that (g) rO .j / D r.j r D 0; : : : ; k C 1, since Zr D fj ¤ 0 W r.j O / D rg, r D 0; : : : ; k C 1.

5.3.3 Asymptotic expansions for solutions of -hitting type systems of linear equations Let us again consider the vector function y."/ and assume that it satisfies the non.0/ negativity condition S1 . For every j 2 X1 consider the system of linear equations, ."/

."/

xj ./ D y."/ C j P."/ ./xj ./:

(5.3.9)

If conditions A2 , E2 , T2 , C17 , C18, and S1 hold, then system (5.3.9) has a unique .0/ solution for every "  "01 and j 2 X1 , ."/

xj ./ D ŒI  j P."/ ./1 y."/ :

(5.3.10) ."/

Also note that, under condition S1 , the components of the vector xj ./ are all non-negative. The following lemma gives vector asymptotic expansions for solutions of these systems. Lemma 5.3.3. Let conditions A2 , E2 , T2 , C17 , C18 , and S1 hold, and also conditions .;k/ .k/ P23 and P19 be satisfied for some   ˇ. Then

296

5

Nonlinearly perturbed semi-Markov processes

."/

.0/

(i) The vector xj ./ has the following asymptotic expansion for every j 2 X1 : ."/

.0/

xj ./ D xj ./ C xj Œ; 1" C    C xj Œ; k"k C o."k /;

(5.3.11)

where xj Œ; r, r D 1; : : : ; k, are finite vectors. (ii) The vector coefficients xj Œ; r are given by the recurrence formulas xj Œ; 0 D .0/ D ŒI  P.0/ ./1 y.0/ and, for r D 0; : : : ; k, x.0/ j j ./ D j UŒ; 0y xj Œ; r D j UŒ; 0.yŒr C

r X

j EŒ; q; 0 xj Œ; r

 q/:

(5.3.12)

qD1

(iii) The vector coefficients xj Œ; r can also be calculated by the following formulas for r D 0; : : : ; k: xj Œ; r D

r X

j UŒ; p yŒr

 p:

(5.3.13)

pD0 .k/

Proof. Relations (5.3.4) and (5.3.5) and condition P19 , expressed in a vector form, imply that conditions of Lemma 8.1.4 are satisfied by the system of nonlinearly perturbed linear equations (5.3.9) for every j 2 X1.0/ . By applying this lemma to systems (5.3.9) we get the claims of Lemma 5.1.1. For r D 1; : : : ; k and j 2 X1.0/ , let 3 2 ."/ ./ x1j 7 6 ."/ :: 7; xj ./ D 6 : 5 4 ."/ xNj ./

3 x1j Œ; r 7 6 :: xj Œ; r D 4 5: : xNj Œ; r 2

Explicit expressions for components of the vectors xŒ; r, r D 0; : : : ; k, can be derived from the recurrence formulas (5.3.12) or formulas (5.3.13). There is a simple probabilistic interpretation of solution of system (5.3.9). In fact, using the probability representation for elements of the matrix ŒI  j P."/ ./1 given by formula (5.3.6), we can write the following representation for components of the ."/ .0/ vector xj ./ for i ¤ 0, j 2 X1 : ."/

xij ./ D

X

."/ ."/ j uil ./yl

l¤0

D

X l¤0

."/

."/

."/

Ei ıj l ./yl

."/

j ^0

D Ei

X

nD1

e 

."/ .n1/

y

."/ ."/

n1

:

(5.3.14)

5.3 Asymptotic expansions for power-exponential moments of hitting times

297

Formula (5.3.14) suggests to refer to systems of type (5.3.9) as -hitting system of linear equations. Hitting systems of type (5.3.9) play an important role in our considerations. Mixed power-exponential moments of hitting times are solutions of such systems. We are especially interested in a study of pivotal properties of asymptotic expan."/ sions for the components xij ./ of the solutions x."/ j ./ of system (5.3.9). A description of such properties is especially important for cyclic components of the vectors ."/ ."/ xj ./, which are the components xjj ./. We will use the sets Yr ; Wr , r D 0; : : : ; k, and conditions O1.h/ , O2.h/ introduced in Subsection 5.1.3. .;k/

Lemma 5.3.4. Let conditions A2 , E2 , T2 , C17 , C18 , and S1 hold, and conditions P23 .k/ and P19 be satisfied for some   ˇ. Then .h/

(i) If condition O1 holds for some 0  h  k, then the first h coefficients are .0/ zeros, i.e., xjj Œ; n D 0, j 2 X1 , n D 0; : : : ; h. .h/

(ii) If condition O2 holds for some 0  h  k, then the first h  1 coefficients .0/ are zeros, i.e., xjj Œ; n D 0, j 2 X1 , n D 0; : : : ; h  1, but xjj Œ; h > 0, .0/ j 2 X1 . Proof. The proof is analogous to the proof of Lemma 5.1.4. Using asymptotic expan.k/ sions (5.3.7) for the -potential matrices j U."/ ./, condition P19 , and Lemma 5.3.2, ."/ .0/ we rewrite relation (5.3.14) for the cyclic components xjj ./, j 2 X1 , in the following form: X ."/ ."/ ."/ xjj ./ D j uj l ./yl l¤0

D

X

X

X

0r k r 0 Cr 00 Dr l2Yr 0 \Zr 00

D

X

X

X

 @

X

0 @

0r k r 0 Cr 00 Dr l2Yr 0 \Zr 00

0

."/ ."/ j uj l yl

1

X

X

C

."/ ."/ j uj l ./yl

r 0 Cr 00 kC1 l2Yr 0 \Zr 00

X

1

p 00

"

j uj l Œ; p

00

A

r 00 p 00 k

0

"p yl Œp 0 A C o."k /

r 0 p 0 k

 X  "r Lj Œ; r; r C    C "k Lj Œ; k; r C o."k / D 0r k

D

X

0r k



"r Lj Œ; r; 0 C    C Lj Œ; r; r C o."k /;

(5.3.15)

298

5

Nonlinearly perturbed semi-Markov processes

where the coefficients Lj Œ; p; r, p D r; : : : ; k, r D 0; : : : ; k, are readily calculated by multiplying the inner sums in (5.3.15) and collecting the coefficients at "p . For every r D 0; : : : ; k, (g) the coefficients Lj Œ; p; r, p D r; : : : ; k, are sums of numbers over the set Wr . In particular, (h) the coefficients Lj Œ; r; r have the following form for r D 0; : : : ; k: X X 00 0 (5.3.16) Lj Œ; r; r D j uj l Œ; r  yl Œr : r 0 Cr 00 Dr l2Yr 0 \Zr 00

Formula (5.3.15) implies that (i) the coefficients xjj Œ; r can be represented for r D 0; : : : ; k as X xjj Œ; r D Lj Œ; r; p: (5.3.17) 0pr .h/

If condition O1 holds for some 0  h  k, then Wr D ¿; r  h. Thus, (j) the sums that define the coefficients LŒ; p; r; p D r; : : : ; k, are actually the sums over the empty set for r  h. Therefore, (k) LŒ; p; r D 0; p D r; : : : ; k, for r  h. The claim (k) and formula (5.1.35) imply that (l) xjj Œ; r D 0; r  h. This proves the first statement of the lemma. .h/ If condition O2 holds for some 0  h  k, then Wr D ¿, r  h  1, but Wh ¤ ¿. This means that condition O1.h1/ holds and, therefore, (m) xjj Œ; r D 0, r  h  1, as was proved above. Moreover, as was shown, (n) LŒ; p; r D 0, p D r; : : : ; k for r  h  1. It is clear that (n) and formula (5.1.35) imply in this case that (o) xjj Œ; h D Lj Œ; r; r. If Wh ¤ ¿, then Yr 0 \ Zr 00 ¤ ¿ for some non-negative integers r 0 and r 00 such that r 0 C r 00 D h. By Lemma 5.3.2 and the definition of the set Zr 00 , we have j uj l Œ; r 00  > . Also, by the definition of the set Yr 0 , and condition S1 , yl Œr 0  > 0, l 2 0, l 2 Zr 00 P Yr 0 . Thus, l2Yr 0 \Zr 00 j uj l Œ; r 00 yl Œr 0  > 0. Therefore, xjj Œ; r D Lj Œ; r; r > 0 according formula (5.3.16). This proves the second statement of the lemma.

5.3.4 Perturbation conditions for mixed power-exponential moments of hitting times We introduce mixed power-exponential moments of transition times, for n D 0; 1; : : :, i; j ¤ 0, and  2 R1 , Z 1 Z 1 ."/ ."/ n s ."/ s e Qij .ds/; s n e s Fij."/ .ds/: pij Œ; n D ij Œ; n D 0

0

The mixed power-exponential moments introduced above are connected by the following formulas, for  2 R1 and i; j ¤ 0, n D 0; 1; : : :, ."/

."/

pij Œ; n D pij

."/ ij Œ; n:

5.3 Asymptotic expansions for power-exponential moments of hitting times

299

Choose an arbitrary 0 < ˇ < ı. The following inequality holds for every n D 1; : : :, i; j ¤ 0: Z 1 Z 1 ."/ ."/ ."/ n ˇs ."/ s e Qij .ds/  cn e ˇ s Qij .ds/ D cn pij .ˇ/; (5.3.18) pij Œˇ; n D 0

0

where cn D cn .ı; ˇ/ D sups0 s n e .ıˇ /s < 1. Thus, condition C17 guarantees that for every 0  ˇ < ı there exists "00 D "00 .ˇ/ > 0 such that, for n D 0; 1; : : :, i; j ¤ 0,   ˇ, and "  "0 , ."/

sup pij Œˇ; n < 1:

(5.3.19)

""00

Conditions T2 and C17 imply that, for n D 0; 1; : : :, j ¤ 0 and   ˇ, we have ."/

.0/

pij Œ; n ! pij Œ; n as " ! 0:

(5.3.20)

We assume that the following perturbation condition holds for some   ˇ: .;k/ P25 :

."/

.0/

pij Œ; n D pij Œ; n C eij Œ; 1; n" C    C eij Œ; k  n; n"kn C o."kn / for n D 0; : : : ; k, i; j ¤ 0, where jeij Œ; r; nj < 1, r D 1; : : : ; k  n, n D 0; : : : ; k, i; j ¤ 0. .0/

It is convenient to define eij Œ; 0; n D pij Œ; n for n D 0; : : : ; k, i; j ¤ 0. ."/

.;k/

Note that moments pi0 Œ; n are not involved in condition P25 . It is because of these moments do not penetrate formulas for mixed power-exponential moments of hitting without absorption times. .k/ In some cases, it will be more convenient to use the perturbation condition P18 ."/ for the transition probabilities pij , i; j ¤ 0, and to additionally use the following perturbation condition imposed on the mixed power-exponential moments of transition ."/ times, ij Œ; n, n D 1; : : : ; k, i; j ¤ 0: .;k/

P26

."/

.0/

: ij Œ; n D ij Œ; n C vij Œ; 1; n" C    C vij Œ; k  n; n"kn C o."kn / for n D 0; : : : ; k, i; j ¤ 0, where jvij Œ; r; nj < 1, r D 1; : : : ; k  n, n D 0; : : : ; k, i; j ¤ 0.

Define vij Œ; 0; n D

.0/ ij Œ; n

for i; j ¤ 0, n D 0; : : : ; k.

.k/ .;k/ .;k/ It can be proved that conditions P18 and P26 imply that condition P25 is veri.k/ .;k/ fied. Moreover, the corresponding asymptotic expansions in conditions P18 and P26 .;k/ are connected with the asymptotic expansion in condition P25 by the following formulas for i; j ¤ 0 and r D 0; : : : ; k  n, n D 0; : : : ; k:

eij Œ; r; n D

r X mD0

eij Œr; 0vij Œ; r  m; n:

(5.3.21)

300

5

Nonlinearly perturbed semi-Markov processes

For n D 0; 1; : : : ; k, consider the matrices 3 ."/ ."/ p11 Œ; n : : : p1N Œ; n 7 6 :: :: :: P."/ Œ; n D 4 5; : : : ."/ ."/ pN1 Œ; n : : : pN N Œ; n 2

and, for r D 0; : : : ; k  n, n D 1; : : : ; k, 2 e11 Œ; r; n 6 :: EŒ; r; n D 4 :

3 e1N Œ; r; n 7 :: 5: : eN1 Œ; r; n : : : eN N Œ; r; n

.;k/

Condition P25

::: :: :

can be rewritten in the following equivalent matrix form:

.;k/

: P."/ Œ; n D P.0/ Œ; n C EŒ; 1; n" C    C EŒ; k  n; n"kn C o."kn /, where P.0/ Œ; n and EŒ; r; n, r D 0; : : : ; k  n, n D 0; : : : ; k are finite matrices. For n D 0; : : : ; k, j ¤ 0, consider the matrices

P25

jP

."/

2

Œ; n D

3 ."/ ."/ ."/ ."/ p11 Œ; n : : : p1.j 1/ Œ; n 0 p1.j C1/ Œ; n : : : p1N Œ; n 6 7 :: :: :: :: :: 6 7; : : : : : 4 5 ."/ ."/ ."/ ."/ pN1 Œ; n : : : pN.j 1/ Œ; n 0 pN.j C1/ Œ; n : : : pN N Œ; n

and also, for r D 0; : : : ; k  n, n D 1; : : : ; k and j ¤ 0, j EŒ; r; n

D

3 e11 Œ; r; n : : : e1.j 1/ Œ; r; n 0 e1.j C1/ Œ; r; n : : : e1N Œ; r; n 7 6 :: :: :: :: :: 5: 4 : : : : : eN1 Œ; r; n : : : eN.j 1/ Œ; r; n 0 eN.j C1/ Œ; r; n : : : eN N Œ; r; n 2

.;k/ implies that for j ¤ 0, the following matrix asymptotic expanCondition P23 sions take place for n D 0; : : : ; k: jP

."/

Œ; n D j P.0/ Œ; n C j EŒ; 1; n" C    C j EŒ; k  n; n"kn C o."kn /: .0/

(5.3.22)

Let eij Œ; 0; n D pij Œ; n, n D 0; : : : ; k, i; j ¤ 0, and, consequently, EŒ; 0; n D .0/ P Œ; n and j EŒ; 0; n D j P.0/ Œ; n, n D 0; : : : ; k, j ¤ 0. We shall also use the notation j P."/ ./ D j P."/ Œ; 0.

5.3 Asymptotic expansions for power-exponential moments of hitting times

301

For r D 0; : : : ; k  n, n D 0; : : : ; k and j ¤ 0, consider the vectors 2 ."/ 3 3 2 p1j Œ; n e1j Œ; r; n 6 7 7 6 ."/ :: :: 7; pj Œ; n D 6 ej Œ; r; n D 4 5: : : 4 5 ."/ eNj Œ; r; n pNj Œ; n .;k/ gives the following vector asymptotic expansion for j ¤ 0: Condition P23 ."/

.0/

pj ./ D pj ./ C ej Œ; 1; 0" C    C ej Œ; k; 0"k C o."k /: .;k/

Condition P25 ."/

(5.3.23)

, for j ¤ 0 and n D 0; : : : ; k, yields .0/

pj Œ; n D pj Œ; n C ej Œ; 1; n" C    C ej Œ; k  n; n"kn C o."kn /:

(5.3.24)

.0/ Œ; n, n D 0; : : : ; k, i; j ¤ 0. Hence, ej Œ; 0; n D Recall that eij Œ; 0; n D pij

."/ ."/ p.0/ j Œ; n, n D 0; : : : ; k, j ¤ 0. We shall also use the notation pj ./ D pj Œ; 0.

It is useful to note that the perturbation conditions P.0;k/ and P.0;k/ 23 25 , in the case .k/ .k/ where  D 0, reduce to more simple conditions P18 and P20 , respectively,

5.3.5 Recursive systems of linear equations for mixed power-exponential moments of hitting times Recall that the moment generating functions for hitting times, introduced in Section 4.5, are given by Z 1 ."/ ."/ ."/ ."/ ."/  ./ D e s 0 Gij .ds/ D Ei e j .j < 0 g;  2 R1 ; i; j ¤ 0; 0 ij 0

and the mixed power-exponential moments for hitting times for n D 0; 1; : : : are Z 1 ."/ ."/ s n e s 0 Gij."/ .ds/ D Ei e j .j."/ < 0."/ g;  2 R1 ; i; j ¤ 0: 0 ij Œ; n D 0

."/

."/

By the definition, 0 ij Œ; 0 D 0 ij ./, i; j ¤ 0. Assume that conditions A2 , E2 , T2 , C17 , and C18 hold. As was mentioned above, Lemma 4.5.6 implies that (a) there exist ˇ < ˇi , where ˇi are given in condition C18 , and 0 < "01 D "1 .ˇ/  "00 .ˇ/ such that relation (5.3.5) .0/ holds and the inverse matrix ŒI  j P."/ ./1 exists for every   ˇ, j 2 X1 and 0 "  "1 .

302

5

Nonlinearly perturbed semi-Markov processes

Lemma 4.5.6 also implies that (b) 0 < ˇ < ˇi can be chosen so that, for every   ˇ, i ¤ 0, j 2 X1.0/ , and "  "01 , we have ."/ 0 ij ./

.0/

! 0 ij ./ < 1 as " ! 0:

(5.3.25)

Moreover, what is also very important for further consideration is that (c) 0 < ˇ < ˇi can be chosen in such way that, for every "  "01 , there exists a unique root ."/  0 of the characteristic equation 0 ij."/ ./ D 1, which (c1 ) does not depend on the choice

of the state j 2 X1.0/ , (c2 ) satisfies the inequality ."/ < ˇ, and is such that (c3 ) ."/ ! .0/ as " ! 0. Relation (5.3.25) implies that (d) there exists 0 < "02 D "02 .ˇ/ < "01 .ˇ/ such that, .0/ for i ¤ 0; j 2 X1 , ."/

sup 0 ij .ˇ/ < 1:

(5.3.26)

""2

Choose an arbitrary .0/ < ˇ 0 < ˇ. Then, for every n D 0; 1; : : :, i ¤ 0, j 2 X1.0/ , ."/ 0 0 ij Œˇ ; n

Z D 

1 0

cn0

s n eˇ

Z

1

0s

."/ 0 Gij .ds/ ."/

."/

e ˇs 0 Gij .ds/ D cn0 0 ij .ˇ/;

0

where cn0 D cn0 .ˇ; ˇ 0 / D sups0 s n e .ˇ ˇ

0 /s

(5.3.27)

< 1.

Relation (5.3.27) implies that (e) for every n D 0; 1; : : :, i ¤ 0, j 2 X1.0/ , ."/

sup 0 ij Œˇ; n < 1:

""2

(5.3.28)

The properties (c1 )–(c3 ) imply that (f) there exists 0 < "03 D "03 .ˇ; ˇ 0 /  "2 .ˇ/ such that sup ."/ < ˇ 0 :

""3

(5.3.29)

Relation (5.3.29) allows us to work with perturbation conditions for mixed powerexponential moments of hitting times for  D .0/ and to get corresponding asymptotic expansions for the roots ."/ . These expansions will be obtained in the next section. To do this, however, there is no need in obtaining asymptotic expansions for the mixed ."/ power-exponential moments 0 jj Œ; n. By using the Markov property of the semi-Markov process ."/ .t / at the moment of the first jump and calculating the corresponding expectation, which depends on the

5.3 Asymptotic expansions for power-exponential moments of hitting times

303

position of this process at the moment of first jump, (f) we obtain the following system of linear equations for every   ˇ 0 , j 2 X1.0/ and "  "3 : ."/ 0 ij ./

X

."/ D pij ./ C

."/ ."/ pil ./0 lj ./;

i ¤ 0:

(5.3.30)

l¤j;0

It follows from relation (5.3.19) that (g) for every n  1 there exists a derivative of ."/ order n of the moment generating function ij ./ and this derivative is the function ."/ ij Œ; n.

This is true for every i; j ¤ 0,   ˇ, and "  "3 . It easily follows from relation (5.3.28) that (h) for every n D 1; : : : the moment ."/ ."/ generating function 'ij ./ has a derivative of order n, and it is the function 'ij Œ; n. .0/

This is true for every i ¤ 0; j 2 X1 ,   ˇ0 , and "  "3 . Using statements (f)–(h), we can differentiate equations (5.3.30) and get the following system of linear equation for every j 2 X1.0/ ,   ˇ0 , "  "3 , and n D 0; 1; : : :: ij."/ Œ; n D ."/ ij Œ; n C

X

."/ ."/ pil ./lj Œ; n;

i ¤ 0;

(5.3.31)

l¤0;j

where ."/

."/

ij Œ; n D pij Œ; n C

n X X

."/

."/

Cnm pil Œ; mlj Œ; n  m:

(5.3.32)

l¤0;j mD1 ."/ ."/ .0/ Note that ."/ ij Œ; 0 D pij Œ; 0 D pij ./, i ¤ 0, j 2 X1 . Thus, system (5.3.31) coincides with system (5.3.30) for n D 0. .0/ These systems can be rewritten in a matrix form. For j 2 X1 and n D 0; : : :, consider the vectors 2 2 ."/ 3 3 ."/ 1j Œ; n 0 1j Œ; n 6 6 7 7 ."/ :: :: 6 7 ; œ."/ Œ; n D 6 7: 0 ¥j Œ; n D 4 : : j 4 5 5 ."/ ."/ Nj Œ; n 0 Nj Œ; n .0/

Then, systems (5.3.30) (see, also, (4.5.3)) can be rewritten for every j 2 X1 ,   ˇ0 , "  "3 , as ."/ 0 ¥j ./

."/

."/

D pj ./ C j P."/ ./ 0 ¥j ./;

(5.3.33)

.0/

and systems (5.3.31) for every j 2 X1 ,   ˇ0 , "  "3 , and n D 0; 1; : : : become ."/ 0 ¥j Œ; n

."/ ."/ D œ."/ j Œ; n C j P ./ 0 ¥j Œ; n:

(5.3.34)

304

5

Nonlinearly perturbed semi-Markov processes ."/

If n D 0, system (5.3.34) coincides with system (5.3.33), since œj Œ; 0 D

."/

.0/

pj ./, j 2 X1 . The systems of linear equations (5.3.34) are systems of -hitting type with the same coefficient matrix j P."/ ./. As was pointed above, conditions A2 , T1 , E2 , C17 , C18 , and S1 imply that there exists "0 > 0 such that for every "  "0 any -hitting type system of linear equation .0/ with the coefficient matrix j P."/ ./ has a unique solution for j 2 X1 ,   ˇ 0 , and "  "3 . Obviously, the vectors œ."/ j Œ; n have non-negative components for every j 2 .0/

X1 ;   ˇ 0 , "  "3 , and n D 0; : : : . In the case of the mixed power-exponential moments of hitting times, this solution .0/ has the following form for j 2 X1 ,   ˇ0 , "  "03 , and n D 0; : : :: ."/ 0 ¥j Œ; n

D ŒI  j P."/ ./1 œ."/ j Œ; n:

(5.3.35)

As was mentioned above, systems (5.3.34), for a given j 2 X1.0/ , have the same ."/ coefficient matrix j P."/ ./ but different inhomogeneous terms œj Œ; n for n D 0; 1; : : : . These systems should be solved recursively. First, the system (5.3.34) should be solved for n D 0. Note that it actually becomes ."/ a system of linear equations for the moment generating functions 0 ij ./, i ¤ 0. Second, we solve system (5.3.34) for n D 1. Note that expressions for the inhomoP ."/ ."/ ."/ ."/ geneous terms ij Œ; 1 D pij Œ; 1 C l¤0;j pil Œ; 10 lj Œ; 0, i ¤ 0, given in

(5.3.32) include the solutions 0 ij."/ Œ; 0, i ¤ 0, of systems (5.3.34) for n D 0. This recursive procedure should be repeated for n D 1; : : : . The expressions for ."/ ."/ the inhomogeneous terms ij Œ; n given in (5.3.32) include the solutions ij Œ; m, i ¤ 0, of systems (5.3.34) for m D 0; 1; : : : ; n  1. .0/ Formula (5.3.32) can be rewritten for every j 2 X1 ,   ˇ 0 , "  "03 , and n D 0; 1; : : : in the following vector form: ."/ œ."/ j Œ; n D pj Œ; n C

n X

Cnm j P."/ Œ; m 0 ¥."/ j Œ; n  m:

(5.3.36)

mD1

Using formula (5.3.36) we can rewrite formula (5.2.10) in a form representing the recursive procedure for calculating solutions of the system of linear equations (5.3.34) for j 2 X1.0/ ,   ˇ0 , "  "03 , and n D 0; 1; : : :, ."/ 0 ¥j Œ; n

D ŒI  j P."/ ./1 œ."/ j Œ; n D ŒI  j P."/ ./1 p."/ j Œ; n C

n X mD1

Cnm ŒI  j P."/ ./1 j P."/ Œ; m 0 ¥."/ j Œ; n  m:

(5.3.37)

5.3 Asymptotic expansions for power-exponential moments of hitting times

305

5.3.6 Asymptotic expansions for power-exponential moments of hitting times The following lemmas are corollaries of Lemma 5.3.4 applied to the vectors of power ."/ .0/ moments of the hitting times, 0 ¥j Œ; n, for j 2 X1 and n D 0; 1; : : : . The next lemma is a direct corollary of Lemma 5.3.4 applied to the vectors of the ."/ .0/ moment generating functions 0 ¥."/ j ./ D 0 ¥j Œ; 0 for j 2 X1 . In Lemmas 5.3.5–5.3.8, .0/ < ˇ 0 < ˇ, where ˇ is defined in (5.3.25). .;k/

Lemma 5.3.5. Let conditions A2 , T2 , E2 , C17 , C18 hold, and condition P23 isfied for some   ˇ0 . Then .0/

be sat-

."/

(i) For every j 2 X1 , the vector 0 ¥j ./ has the following asymptotic expansion: ."/ 0 ¥j ./

.0/

D 0 ¥j ./ C 0 bj Œ; 1; 0" C    C 0 bj Œ; k; 0"k C o."k /; (5.3.38)

where 0 bj Œ; r; 0, r D 1; : : : ; k, are finite vectors. (ii) The vector coefficients 0 bj Œ; r; 0 are given by the recurrence formulas .0/ .0/ .0/ 1 .0/ 0 bj Œ; 0; 0 D 0 ¥j ./ D j UŒ; 0pj ./ D ŒI  j P ./ pj ./ and, consequently, for r D 0; : : : ; k, 0 bj Œ; r; 0 D j UŒ; 0.ej Œ; r; 0 C

r X

j EŒ; q; 00 bj Œ; r

 q; 0/:

(5.3.39)

qD1

(iii) The vector coefficients 0 bj Œ; r; 0 can also be calculated by the following formulas for r D 0; : : : ; k: 0 bj Œ; r; 0

D

r X

j UŒ; p ej Œ; r

 p; 0:

(5.3.40)

pD0

Proof. We apply Lemma 5.3.4 to the system of linear equations (5.3.34) for every ."/ j 2 X1.0/ and n D 0. Here, the vector p."/ j ./ is regarded as the vector y . The non-negativity condition S1 obviously holds. Also, the asymptotic relations (5.3.23) .0/ .k/ imply that, for every j 2 X1 , condition P19 holds. Applying Lemma 5.3.4 to the system of linear equations (5.3.34) we finish the proof of Lemma 5.3.5. The following lemma is a direct corollary of Lemma 5.3.4 applied to the vectors of ."/ .0/ mixed power-exponential moments 0 ¥j Œ; n, n D 0; 1; : : : ; k for j 2 X1 . .;k/

Lemma 5.3.6. Let conditions A2 , T2 , E2 , C17 , C18 hold, and condition P25 isfied for some   ˇ0 . Then

be sat-

306

5

Nonlinearly perturbed semi-Markov processes .0/

."/

(i) For every j 2 X1 and n D 0; : : : ; k, the vector 0 ¥j Œ; n has the following asymptotic expansion: ."/ 0 ¥j Œ; n

.0/

D 0 ¥j Œ; n C 0 bj Œ; 1; n" C    C 0 bj Œ; k  n; n"kn C o."kn /;

(5.3.41)

where 0 bj Œ; r; n, r D 1; : : : ; k  n, n D 0; : : : ; k, are finite vectors. (ii) The vector coefficients 0 bj Œ; r; n are given by the recurrence formulas .0/ .0/ .0/ 1 .0/ 0 bj Œ; 0; 0 D 0 ¥j Œ; 0 D j UŒ; 0œj Œ; 0 D ŒI  j P ./ pj ./ for r D 0; : : : ; k  n and fixed n and sequentially for, n D 0; : : : ; k, 0 bj Œ; r; n

D j UŒ; 0.ej Œ; r; n C

n X

Cnm

mD1

C

r X

r X

j EŒ; q; m0 bj Œ; r

 q; n  m

qD0

Ej Œ; p; 0 0 bj Œ; r  p; n/:

(5.3.42)

pD1

(iii) The vector coefficients 0 bj Œ; r; n can also be calculated by the recurrence formulas for r D 0; : : : ; k  n for given and fixed n and sequentially for, n D 0; : : : ; k, 0 bj Œ; r; n

D

r X

j UŒ; qej Œ; r

qD0

C

n X mD1

Cnm

r X qD0

 q; n

0 @

q X

1 j UŒ; pj EŒ; q

 p; mA

pD0

 0 bj Œ; r  q; n  m:

(5.3.43)

.0/

Proof. Fix some j 2 X1 . The asymptotic expansions (5.3.41) constructed for the ."/ vectors 0 ¥j Œ; n are obtained recursively for n D 0; : : : ; k. For n D 0, we set 0 œ."/ j Œ0 D .;k/ P25

."/ 0 pj ./.

The asymptotic expansion for n D 0 in .;k/

condition coincides with the asymptotic expansion given in condition P23 . ."/ The asymptotic expansion (5.3.41) for the vectors 0 ¥j Œ; 0 and formulas (5.3.42) and (5.3.43) for the corresponding vector coefficients coincide, respectively, with the ."/ asymptotic expansion (5.3.38) for the vectors 0 ¥j ./ and formulas (5.3.39) and (5.3.40) for the corresponding vector coefficients. Note that this is an .N; 1/-matrix .0; k/-expansion.

5.3 Asymptotic expansions for power-exponential moments of hitting times ."/

307

."/

Now we can write an asymptotic expansion for the vectors œj Œ; 1 D pj Œ; 1 C jP

."/ Œ; 1

."/ 0 ¥j Œ; 0

using Lemma 8.1.2. This will be an .N; 1/-matrix .0; k  1/-ex-

."/ 1  pansion. Then, Lemma 5.1.3 can be applied to the vectors 0 ¥."/ j Œ1 D ŒI  j P ./ ."/

œj Œ; 1. The resulted asymptotic expansion is an .N; 1/-matrix .0; k 1/-expansion. The recurrence procedure is repeated for n D 1; : : : ; k. At step n, the expansion for the vectors œ."/ j Œ; n given by formula (5.2.11) is an .N; 1/-matrix .0; kn/-expansion that takes the following form: .0/ œ."/ j Œ; n D œj Œ; n C œj Œ; 1; n" C   

C œj Œ; k  n; n"kn C o."kn /;

(5.3.44)

where œ.0/ j Œ; n D œj Œ; 0; n and the coefficients œj Œ; r; n, r D 0; : : : ; k  n, are given by the formulas œj Œ; r; n D ej Œ; r; n C

n X mD1

Cnm

r X

j EŒ; q; m0 bj Œ; r

 q; n  m:

(5.3.45)

qD0

."/ 1 ."/ Now, apply Lemma 5.3.4 to the vectors 0 ¥."/ j Œ; n D ŒI  j P ./ œj Œ; n. The resulted asymptotic expansion is an .N; 1/-matrix .0; k  n/-expansion given by asymptotic expansion (5.3.41) and formulas (5.3.42) and (5.3.43) for the corresponding vector coefficients.

Remark 5.3.1. It should be noted that both formulas (5.3.42) and (5.3.43) for the corresponding vector coefficients in the asymptotic expansions (5.3.41) have a recursive character. However, they differ in their structure. For calculating the coefficients 0 bj Œ; r; n in (5.3.42) we use the coefficients 0 bj Œ; l; m, l D 0; : : : ; r, m D 0; : : : ; n  1. However, in formulas (5.3.43) the coefficients 0 bj Œ; r; n are calculated from the coefficients 0 bj Œ; l; m, l D 0; : : : ; r, m D 0; : : : ; n  1, and also the coefficients 0 bj Œ; l; n, l D 0; : : : ; r  1. Take integer parameters nQ  1 and kQn  0; n D 0; : : : ; n, Q and define a vector parameter kQ D .n; Q kQ0 ; : : : ; kQnQ /. Assume that the following perturbation condition, which is more general than .;k/ P25 , holds for some   ˇ 0 : .;kQ /

P27

Q Q ."/ .0/ : pij Œ; n D pij Œ; n C eij Œ; 1; n" C    C eij Œ; kQn ; n"kn C o."kn / for .0/

Q i; j ¤ 0, and n D 0; : : : ; n, Q i; j ¤ 0, where pij Œ; n < 1, n D 0; : : : ; n, Q jeij Œ; r; nj < 1, r D 1; : : : ; kn , n D 0; : : : ; n, Q i; j ¤ 0. .;kQ / .;k/ Condition P27 reduces to condition P25 if nQ D k; kQn D k  n, n D 0; : : : ; k. The following lemma generalises Lemma 5.2.1 and reduces to this lemma in the case mentioned above.

308

5

Nonlinearly perturbed semi-Markov processes .;k/

Lemma 5.3.7. Let conditions A2 , T2 , E2 , C17 , C18 hold, and condition P27 isfied for some   ˇ0 . Then

be sat-

(i) The vector 0 ¥."/ j Œ; n has the following asymptotic expansion for every j 2 .0/

X1

and n D 0; : : : ; n: Q ."/ 0 ¥j Œ; n

D 0 ¥.0/ j Œ; n C 0 bj Œ; 1; n" C    C 0 bj Œ; kn ; n"kn C o."kn /;

(5.3.46)

where kn D min.kQ0 ; : : : ; kQn /, n D 0; : : : ; n, Q and 0 bj Œ; r; n, r D 1; : : : ; kn , n D 0; : : : ; n, Q are finite vectors. (ii) The vector coefficients 0 bj Œ; r; n are given by the recurrence formulas .0/ .0/ .0/ 1 .0/ 0 bj Œ; 0; 0 D 0 ¥j Œ; 0 D j UŒ; 0œj Œ; 0 D ŒI  j P ./ pj ./ for Q r D 0; : : : ; kn and sequentially for n D 0; : : : ; n, 0 bj Œ; r; n

D j UŒ; 0.ej Œ; r; n C

n X

Cnm

mD1

C

r X

r X

j EŒ; q; m0 bj Œ; r

 q; n  m

qD0

Ej Œ; p; 0 0 bj Œ; r  p; n/:

(5.3.47)

pD1

(iii) The vector coefficients 0 bj Œ; r; n are also calculated by the recurrence formulas for r D 0; : : : ; kn and sequentially for n D 0; : : : ; n, Q 0 bj Œ; r; n

D

r X

j UŒ; qej Œ; r

qD0

C

n X mD1

Cnm

r X qD0

 q; n

0 @

q X

1 j UŒ; pj EŒ; q

 p; mA

pD0

 0 bj Œ; r  q; n  m:

(5.3.48)

Proof. The proof repeats the proof of Lemma 5.3.6. Lemma 5.3.7 differs from Lemma 5.3.6 in the orders of the corresponding asymptotic expansions for the power moments of the hitting times 0 ¥."/ j Œ; n. These orders also should be calculated recursively. ."/

As was mentioned above, 0 ¥j Œ; 0 D .;kQ / P27

."/ 0 ¥j ./

for n D 0. The asymptotic

expansion in condition given for n D 0 coincides with the asymptotic expansion .;k/ in condition P23 if we set k D kQ0 . The asymptotic expansion (5.3.46) for the

5.3 Asymptotic expansions for power-exponential moments of hitting times ."/

309

."/

vectors 0 mj Œ; 0 D ŒI  j P."/ ./1 œj Œ; 0 and formulas (5.3.47) and (5.3.48) for the corresponding vector coefficients just coincide, respectively, with the asymptotic ."/ expansion (5.3.38) for the vectors 0 ¥."/ j ./ D 0 ¥j Œ; 0 and formulas (5.3.39) and (5.3.40) for the corresponding vector coefficients. Note that this is an .N; 1/-matrix .0; k0 /-expansion, where k0 D kQ0 . Indeed, according to Lemma 8.1.4, the order of this expansion is the minimum of orders of the asymptotic expansions for the matrices ."/ .;k/ ."/ j P ./ and the vectors œj Œ; 0. Both these expansions, by condition P23 , have order k D kQ0 . ."/

."/

Now we can write an asymptotic expansion for the vectors œj Œ; 1 D pj Œ; 1C ."/

."/

Œ; 1 0 ¥j Œ; 0 using Lemma 8.1.2. This is an .N; 1/-matrix .0; k1 /-expansion, where k1 D min.kQ0 ; kQ1 /. Indeed, by Lemma 5.1.3, the order of this expansion is the jP

."/

minimum of orders of the asymptotic expansions for the vectors pj Œ; 1, matrices ."/ ."/ Q Q Q j P Œ; 1, and the vectors 0 ¥ Œ; 0 that are, respectively, k1 , k1 , and k0 . This minj

."/ imum is k1 D min.kQ0 ; kQ1 /. Then, apply Lemma 5.1.3 to the vectors 0 ¥j Œ; 1 D ."/

ŒI  j P."/ ./1 œj Œ; 1. The obtained asymptotic expansion is an .N; 1/-matrix .0; k1 /-expansion given in formula (5.3.46) for n D 1. According to Lemma 8.1.4, the order of this expansion is the minimum of orders of the asymptotic expansions for the matrices j P."/ ./ and the vectors œ."/ j Œ; 1. These expansions have, respectively, the order kQ1 and k1 D min.kQ0 ; kQ1 /. Their minimum is k1 . This recursive procedure can be repeated for n D 1; : : : ; nQ that will yield the corresponding asymptotic expansions given in (5.3.46).

5.3.7 Pivotal properties of asymptotic expansions for mixed power-exponential moments of hitting times For vectors 0 bj Œr; n, r D 0; : : : ; k  n, n D 0; : : : ; k, we denote 2 0 bj Œ; r; n

6 D4

0 b1j Œ; r; n

3

7 :: 5: : 0 bNj Œ; r; n

Explicit expressions for components of the vectors 0 bj Œr; n, r D 0; : : : ; k  n, n D 0; : : : ; k, can be derived from recurrence formulas (5.3.47) and (5.3.48), respectively. Pivotal properties of asymptotic cyclic mixed power-exponential moments ."/ .0/ Q j 2 X1 are simple. 0 jj Œ; n, n D 0; : : : ; n, .;k/ be Lemma 5.3.8. Let conditions A2 , T2 , E2 , I4 , C17 , C18 hold, and condition P27 0 satisfied for some   ˇ . Then the asymptotic expansions for the power moments ."/ .0/ of hitting probabilities 0 jj Œ; n, n D 0; : : : ; k, j 2 X1 , are all 0-pivotal, i.e.,

310

5

Nonlinearly perturbed semi-Markov processes .0/

the corresponding zero coefficients are 0 jj Œ; n D 0 bjj Œ; 0; n > 0 for all n D .0/

0; : : : ; nQ and j 2 X1 . .0/

Proof. Condition I4 implies that (a) there exists a state i 0 2 X1 and j 0 ¤ 0 such that .0/ .0/ .0/ (b) pi 0 j 0 > 0 and (c) 1  Fi 0 j 0 .0/ > 0. Choose an arbitrary state j 2 X1 . There are two cases to be considered. The first case is where i 0 D j . Here, (b) and (c) imply that .0/ .0/ (d) 1 0 Gjj .0/ > pi 0 j 0 .1Fi 0 j 0 .0// > 0. The second case is where i 0 ¤ j . There, (e) .0/

the cyclic hitting probability is 0jfj i 0 > 0, since i 0 is a recurrent-without-absorption .0/ .0/

.0/

state. Using (b), (c), and (e) we get that (f) 1  0 Gjj .0/ > 0jfj i 0 pi 0 j 0 .1  Fi 0 j 0 .0// > .0/ Œ; n > 0, n D 0; : : : ; n. Q 0. Obviously, (d) as well as (f) implies that (i) 0 jj

Remark 5.3.2. Condition I4 is not required to be imposed on the moment generating ."/ ."/ .0/ functions 0 jj Œ; 0 D 0 jj ./ in the corresponding proposition, since 0 jj Œ; 0 > 0 by the definition. The question about “higher” order pivotal properties of mixed power-exponential moments of hitting times is not simple. We are especially interested in such properties ."/ ."/ for the moment generating functions 0 jj ./ D 0 jj Œ; 0, since these assumptions are involved in the conditions of theorems about asymptotic exponential expansions in mixed ergodic and large deviation theorems for perturbed semi-Markov processes. It would be nice to be able to formulate, for the moment generating functions ."/ 0 jj ./, a statement about higher order coefficients in the corresponding asymptotic ."/

expansions similar to those formulated for the hitting probabilities 0 fjj in Lemma 5.1.9. The proof of Lemma 5.1.9 is based on the use of the formula (a) 0 fjj."/ D 1  jfj."/ 0

which connects the hitting probabilities 0 fjj."/ and the absorption probabilities jfj."/ 0 . This formula shows that the higher order coefficients in the asymptotic expansions for ."/ the hitting probabilities 0 fjj equal zero if the corresponding higher order coefficients ."/

in the asymptotic expansions for the absorption probabilities jfj 0 equal zero. Unfortunately, an analogous formula that connects the moment generating functions ."/ ."/ ."/ ."/ ."/ 0 jj ./ and j j 0 ./ has not such a simple form, (b) 0 jj ./ D jj ./  j j 0 ./, ."/

."/ ./ D Ej e .j where jj

."/

^0 /

."/

0 . ."/ <  ."/ /. and j j."/ 0 0 ./ D Ej e j

It is possible to prove, for the moment generation functions j j."/ 0 ./, statements

similar to those formulated in Lemma 5.1.7 on the absorption probabilities jfj."/ 0 , if

.;k/ by assuming that the asymptotic expanwe extend the perturbation condition P23 sion given in this condition also holds for i ¤ 0, j D 0. However, this would not yield a desirable form of an asymptotic expansion for the moment generating func."/ ."/ tions 0 jj ./ similar to those given for the hitting probabilities 0 fjj in Lemma 5.1.9.

311

5.3 Asymptotic expansions for power-exponential moments of hitting times ."/

This is so because the moment generating functions jj ./ ¤ 1 and the coefficients of asymptotic expansions for these functions can also contribute to higher order co."/ ./ even if the corresponding higher efficients in the asymptotic expansions for 0 jj

order coefficients of j j."/ 0 ./ would vanish. For this reason, we can not formulate conditions which would guarantee vanishing the higher order coefficients in the asymptotic expansions for the moment generating ."/ ."/ functions 0 jj ./ in such an explicit form as for the hitting probabilities 0 fjj in .h/

.h/

Lemma 5.1.9. Recall that these conditions are O3 and O4 that are given in terms of the coefficients of the initial perturbation condition for the transition probabilities ."/ . pij We will replace these condition with a less explicit condition that imposes a “zero” assumption directly on the coefficients in the corresponding expansions for the mo."/ ment generating functions 0 jj ./. These conditions for 1  h  k are the following: .;h/

O5

.0/

: bjj Œ; r; 0 D 0, r D 1; : : : ; h for some j 2 X1 ;

and .;h/

O6

.0/

: bjj Œ; r; 0 D 0, r D 1; : : : ; h  1 but bjj Œ; h; 0 > 0 for some j 2 X1 .

The following lemma describes a corresponding solidarity property connected with .;h/ .;h/ and O6 . conditions O5 .;k/

Note that the perturbation condition P27 is used in this lemma for the value  D .0/ .0/ which is the root of the equation 0 jj ./ D 1. ..0/ ;k/

Lemma 5.3.9. Let conditions A2 , T2 , E2 , C17 , C18 hold, and condition P27 satisfied. Then

be

..0/ ;h/

(i) If condition O5 holds for some 1  h  k, then bjj Œ.0/ ; n; 0 D 0, .0/ n D 1; : : : ; h for every j 2 X1 . ..0/ ;h/

(ii) If condition O6 holds for some 1  h  k, then bjj Œ.0/ ; n; 0 D 0, .0/ n D 0; : : : ; h  1 but bjj Œ.0/ ; h; 0 > 0 for every j 2 X1 . Proof. The statement of the lemma is trivial if the set X1.0/ contains only one state. Thus, we assume that this set contains at least two states. Let j 2 X1.0/ be a state for ..0/ ;h/

which condition O5

..0/ ;h/

or O6

.0/

holds, and i ¤ j , i 2 X1 . ."/

It follows from relation (4.5.27) that (g) .1  0 i i .//.1 

."/ 0 jj .//.1

."/ 0j i i .//.

."/ 0i jj .//

D .1 

 As was shown in the proof of Lemma 4.5.3, this formula is valid for any i ¤ j , i; j ¤ 0,  2 R1 .

312

5

Nonlinearly perturbed semi-Markov processes .0/

Apply this formula to  D .0/ . As, as was mentioned above, (h) 0 jj ..0/ // D 1, .0/

."/

."/

."/

j 2 X1 . Also, 0i jj ..0/ //, 0j i i ..0/ / < 1 for any i ¤ j , i; j 2 X1 . It follows .0/

."/

from Lemma 4.2.2 that (i) there exists "0 > 0 such that X1 X1 for "  "0 . Using (g)–(i) we can rewrite formula (4.5.27) for "  "0 in the following form: ."/ .0/ .0/ .0 i.0/ i . /  0 i i . // .0/

."/

D .0 jj ..0/ /  0 jj ..0/ // As follows from Lemma 4.2.5, (j) ."/ .0/ 0j i i . /

!

.0/ .0/ / 0j i i .

.1 

."/ .0/ 0i jj . //

If condition O. 5

.0/ ;h/

.1 

."/ .0/ 0i jj . /

."/ .0/ // 0j i i . : ."/ .0/ // 0i jj .

!

as " ! 0 for i ¤ j; i; j 2

."/

.1  0j i i ..0/ //

.1 

!

.1  .1 

.0/ .0/ 0j i i . // .0/ .0/ 0i jj . //

.0/ .0/ 0i jj . / X1.0/ . Thus,

(5.3.49) as " ! 0 and

> 0 as " ! 0:

(5.3.50)

holds then, by the Lemma 5.3.7, .0/

."/

"h .0 jj ..0/ /  0 jj ..0/ // ! 0 as " ! 0:

(5.3.51)

Relations (5.3.49), (5.3.50), and (5.3.51) imply that, for i ¤ j , i; j 2 X1.0/ , ."/ .0/ .0/ "h .0 i.0/ i . /  0 i i . // ! 0 as " ! 0:

(5.3.52)

Relation (5.3.52) obviously implies that (k) 0 bi i Œ.0/ ; n; 0 D 0, n D 1; : : : ; h, for i ¤ j , i; j 2 X1.0/ . Claim (k) proves the first statement of the lemma. ..0/ ;h/

If condition O6 .0/

holds, then by Lemma 5.3.7, ."/

"h .0 jj ..0/ /  0 jj ..0/ // ! 0 bjj Œ.0/ ; h; 0 > 0 as " ! 0:

(5.3.53) .0/

Relations (5.3.49), (5.3.53), and (5.3.51) imply that, for i ¤ j , i; j 2 X1 , ."/ .0/ .0/ "h .0 i.0/ // i . /  0 i i .

! 0 bjj Œ.0/ ; h; 0

.1  .1 

.0/ .0/ 0j i i . // .0/ .0/ // 0i jj .

> 0 as " ! 0:

(5.3.54)

Relation (5.3.54) obviously implies that (l) 0 bi i Œ.0/ ; n; 0 D 0, n D 1; : : : ; h  1, .0/ .0/ for i ¤ j; i; j 2 X1 and also that (m) 0 bi i Œ.0/ ; h; 0 > 0 for i ¤ j , i; j 2 X1 . Now we use claims (l) and (m) to prove the second statement of the lemma.

5.4 Exponential expansions for nonlinearly perturbed semi-Markov processes

313

5.4 Exponential expansions for nonlinearly perturbed semiMarkov processes In this section we give pseudo-stationary exponential asymptotic expansions in mixed ergodic and large deviation theorems for perturbed semi-Markov processes.

5.4.1 Imbedding in the model of perturbed regenerative processes We will be interested in obtaining mixed limit and large deviation theorems for the ."/ ."/ transition probabilities Pil .t / D Pi f."/ .t / D l; 0 > t g. The process ."/ .t /, t  0, is a regenerative process with absorption, and the times ."/ of return to the initial state are the corresponding regeneration times. Also, 0 is a ."/ regenerative stopping time which regenerates together with the process  .t /, t  0, at the times of return. Therefore, we can use theorems given in Chapter 3. The corresponding imbedding procedure is described in Section 4.2. The basic renewal equation for the case where the corresponding imbedded regenerative process has not a transition period is given in relation (4.2.3), and it takes the .0/ following form for j 2 X1 , l ¤ 0: ."/ Pj l .t /

where

D

."/ qj l .t / C

."/

Z

t 0

."/

."/

Pj l .t  s/ 0 Gjj .ds/; ."/

qj l .t / D Pj f."/ .t / D l; j

."/

> t; 0 > t g;

t  0;

(5.4.1)

t  0:

."/

The distribution of the hitting (return) time 0 Gjj .t / plays the role of the distribution of regeneration times that generates this renewal equation. A crucial role is played by the root of the characteristic equation, Z

1 0

."/

e s 0 Gjj .ds/ D 1:

(5.4.2)

Theorems that we obtain below improve the corresponding mixed limit and large deviation theorems given in Chapter 4, since we construct asymptotic expansions for the roots ."/ of equation (5.4.2). These asymptotic expansions are based on corresponding asymptotic expansions for power and mixed power-exponential moments of the distributions of regeneration times. Here, we will use the asymptotic expansions for such moments constructed for the hitting times in Sections 5.1–5.3. We shall also use the following condition (introduced in Subsection 3.4.2) that balances the rate of growth of time and the rate of the perturbation: ."/ ! 1 as " ! 0 such that "r t ."/ !  , where 0   < 1. B.r/ r r 5 : 0t

314

5

Nonlinearly perturbed semi-Markov processes

5.4.2 Pseudo-stationary exponential expansions in mixed ergodic and large deviation theorems In this section we present pseudo-stationary exponential asymptotic expansions in mixed ergodic and large deviation theorems for perturbed semi-Markov processes. First of all note that the pseudo-stationary asymptotics appear in the case where .0/ the absorption from the states i 2 X1 is asymptotically impossible. This is the case .0/ .0/ where condition O.0/ 3 holds, i.e., pi0 D 0 for every i 2 X1 . .0/

.k/

Theorem 5.4.1. Let conditions A2 , T1 , E2 , C12 , N1 , O3 , and P20 hold. Then (i) For all " small enough there exists a unique nonnegative root ."/ of equation (5.4.2), it does not depend on the choice of the state j 2 X1.0/ , and has the following asymptotic Taylor’s expansion: ."/ D a1 " C    C ak "k C o."k /;

(5.4.3)

where the coefficients an , n D 1; : : : ; k, do not depend on the choice of state .0/ j 2 X1 and are given by the recurrence formulas a1 D bjj Œ1; 0=bjj Œ0; 1 for n D 1; : : : ; k,  n1 X an D  bjj Œ0; 11 bjj Œn; 0 C bjj Œn  q; 1aq qD1

C

n X n X

X

bjj Œn  q; m 

mD2 qDm

q1 Y



n app =np Š

;

(5.4.4)

n1 ;:::;nq1 2Dm;q pD1

where Dm;q , for every 2  m  q < 1, is the set of all nonnegative, integer solutions of the system n1 C    C nq1 D m; n1 C    C .q  1/nq1 D q

(5.4.5) .0/

and the coefficients bjj Œr; n, r D 0; : : : ; kn, n D 0; : : : ; k, for every j 2 X1 , are given by the recurrence formulas (5.2.14) or (5.2.15) in Lemma 5.2.1. .h/

.h/

(ii) If condition O3 holds for some 1  h  k, then a1 ; : : : ; ah D 0. If O4 holds for some 1  h  k, then a1 ; : : : ; ah1 D 0 but ah > 0. (iii) If condition B.r/ 5 holds for some 1  r  k, then the following asymptotic relation holds for i; l ¤ 0: ."/

Pi f."/ .t ."/ / D l; 0 > t ."/ g expf.a1 " C    C ar 1 "r1 /t ."/ g .0/

.0/

! .1  X .0/ fi0 /l e r ar as " ! 0: 1

(5.4.6)

5.4 Exponential expansions for nonlinearly perturbed semi-Markov processes

315

.0/

Remark 5.4.1. In asymptotic relation (5.4.6), l , l ¤ 0, is the stationary distribution for the semi-Markov process .0/ .t /, t  0, given in formula (4.4.14), and f .0/ is the absorption probability given in formula (4.2.77) with the set Z D X1.0/ . X .0/ i0 1

Remark 5.4.2. If condition E2 is realised in the form of condition E02 , then X1.0/ D X0 D f1; : : : ; N g, and the stationary distribution l.0/ , l ¤ 0, is given by a simpler

formula (4.4.7). In this case, we also have that X .0/ fi0.0/ D 0, i ¤ 0. 1

Proof. The proof is based on a use of the corresponding Theorems 3.4.2 and 3.4.7 for perturbed regenerative processes or on the improvements of the corresponding asymptotic relations in Theorems 4.6.1 and 4.6.2 for perturbed semi-Markov processes. The latter theorems yield the following asymptotic relation for i; l ¤ 0 under conditions A2 , T1 , E2 , C13 , N1 , and O.0/ 3 : ."/ g Pi f."/ .t ."/ / D l; ."/ 0 >t ! .1  expf."/ t ."/ g

.0/

X1

fi0.0/ /l.0/ as " ! 0:

(5.4.7)

We should add to this asymptotic relation the asymptotic expansion (5.4.3) for the root ."/ of the characteristic equation (5.4.2). This can be done by using Theorem ."/ 2.1.2. The distribution function 0 Gjj .t / plays the role of the distribution function F ."/ .t / in the characteristic equation (2.1.4). Note that, according to Lemma 4.5.6, the root ."/ does not depend on the choice of the state j 2 X1.0/ . Lemmas 4.2.3, 4.5.6, and 4.5.8 imply that, if conditions A2 , T1 , E2 , C13 , N1 , and .0/ ."/ O3 hold, then the distribution function 0 Gjj .t / satisfies conditions D6 and C1 . Lem.0/

mas 5.1.6, 5.1.9, and 5.2.1 also imply that, if conditions A2 , T1 , E2 , C12 , O3 , and .k/ .k/ P20 hold, the perturbation condition P1 is also satisfied. The hitting probabilities ."/ ."/ 0 fjj and the power moments of hitting times 0 Mjj Œr, r D 1; : : : ; k, replace in this ."/

.k/

case the probabilities 1  f ."/ and the moments mr , r D 1; : : : ; k, in condition P1 . The asymptotic expansion (2.1.25) takes in this case the form of the asymptotic expansion (5.4.3). This proves statement (i) of the theorem. Lemmas 5.1.7 and 5.1.9 imply claim (ii). By substituting the asymptotic expansion (5.4.3) in the asymptotic relation (5.4.7), we transform this relation into the following form: ."/

Pi f."/ .t ."/ / D l; 0 > t ."/ g .0/ .0/ ! .1  X .0/ fi0 /l as " ! 0: (5.4.8) 1 expf.a1 " C    C ak "k C o."k //t ."/ g r ."/ g If the balancing condition B.r/ 5 holds for some 1  r  k, then (a) expfar " t r C1 k k ."/ ! expfar r g as " ! 0 and (b) expf.ar C1 " C  Cak " Co." //t g ! 1 as " ! 0. Using (a) and (b) we can rewrite relation (5.4.8) in the desirable form (5.4.6). This proves assertion (iii) of the theorem.

316

5

Nonlinearly perturbed semi-Markov processes

Introduce the integer parameters h  1, k0  h, nQ D n.h/ Q D max.n W nh  k0 /, Q D .n.h/, Q k0 ; : : : ; kn.h/ /. and k0  k1      knQ  1. Let us also denote k.h/ Q .h1/

Theorem 5.4.2. Let conditions A2 , T1 , E2 , C12 , N1 , O3

.kQ .h//

, and P20

hold. Then

(i) For all " small enough, equation (5.4.2) has a unique nonnegative root ."/ , which does not depend on the choice of the state j 2 X1.0/ and has the following asymptotic expansion: ."/ D ah " C    C ak "k C o."k /;

(5.4.9)

Chn.h//, Q the coefficients an , n D h; : : : ; k where k D min.k0 ; k1 Ch; : : : ; kn.h/ Q .0/ do not depend on the choice of state j 2 X1 and are given by the recurrence formulas ah D bjj Œh; 0=bjj Œ0; 1 and, for n D h; : : : ; k,

an D  bjj Œ0; 1

1

 bjj Œn; 0 C

n1 X

bjj Œn  q; 1aq

qDh

C

Œn= Xh

n X

X

bjj Œn  q; m 

mD2 qDmh

q1 Y

 n app =np Š ; (5.4.10)

n1 ;:::;nq1 2Dh;m;q pD1

where Dh;m;q , for every h  1 and 2  m  q < 1, is the set of all nonnegative, integer solutions of the system nh C    C nq1 D m;

hnh C    C .q  1/nq1 D q;

(5.4.11)

Q are given by the and the coefficients bjj Œr; n, r D 0; : : : ; kn , n D 0; : : : ; n.h/ recurrence formulas (5.2.19) or (5.2.20) in Lemma 5.2.2 for every j 2 X1.0/ . .hQ / (ii) If condition O3 holds for some h  hQ  k, then ah ; : : : ; ahQ D 0. If condition .hQ / O holds for some h  hQ  k, then ah ; : : : ; a Q D 0 but a Q > 0. h1

4

h

.r/

(iii) If condition B5 holds for some 1  r  k, then the following asymptotic relation holds for i; l ¤ 0: ."/

Pi f."/ .t ."/ / D l; 0 > t ."/ g expf.ah "h C    C ar 1 "r1 /t ."/ g .0/

.0/

! .1  X .0/ fi0 /l e r ar as " ! 0: 1

(5.4.12)

5.4 Exponential expansions for nonlinearly perturbed semi-Markov processes

317

Proof. The proof is based on using of the corresponding Theorems 3.4.3 and 3.4.8 for perturbed regenerative processes and on the improvement of the corresponding asymptotic relations given in Theorems 4.6.1 and 4.6.2 for perturbed semi-Markov processes. ."/ Again, the distribution function 0 Gjj .t / plays in this case the role of the distribu."/ tion function F .t / in the characteristic equation (2.1.4). The only difference with the proof of Theorem 5.4.1 is that the corresponding asymptotic expansion for the characteristic root ."/ of the characteristic equation (5.4.2) takes another form with the first h  1 coefficients equal zero. This can be done by using Theorem 2.1.3 instead of Theorem 2.1.2. Lemmas 5.1.6, 5.1.9, and 5.2.1 imply that, under conditions A2 , T1 , Q .kQ .h// , the perturbation condition P.2k.h// holds. The asymptotic exE2 , C12 , O.0/ 3 , and P20 pansion (2.1.36) takes in this case the form of the asymptotic expansion (5.4.9). This proves statement (i) of the theorem. The proof of statements (ii) and (iii) is analogous to those given in Theorem 5.4.1.

5.4.3 Quasi-stationary exponential expansions in mixed ergodic and large deviation theorems In this section, we obtain quasi-stationary exponential asymptotic expansions in mixed ergodic and large deviation theorems for perturbed semi-Markov processes. Note that quasi-stationary asymptotics appear in the case where the absorption from .0/ .0/ the states i 2 X1 is asymptotically possible. This is the case where condition O4 .0/ .0/ holds, that is, pi0 ¤ 0 for some i 2 X1 . .0/

It should be noted that the theorems formulated below do not require condition O4 to hold and cover both pseudo- and quasi-stationary cases. However, in the pseudostationary case, i.e., if condition O.0/ 3 holds, Theorems 5.4.1 and 5.4.2 give more effective conditions for obtaining the corresponding asymptotic relations, in particular, those related to pivotal properties of these expansions. In the quasi-stationary case, the pivotal properties of the asymptotic expansions ."/ for the moment generating functions 0 jj can not be expressed so explicitly via the coefficients in the corresponding asymptotic expansion given in the initial perturba.h/ .h/ tion conditions as it is done for the pseudo-stationary case. Conditions O3 and O4 should be replaced with the less explicit but more general conditions O5.

.0/ ;h/

and

..0/ ;h/ O6 .

..0/ ;k/

Theorem 5.4.3. Let conditions A2 , T1 , E2 , C12 , C13 , N1 , and P25

hold. Then

(i) For all " small enough, equation (5.4.2) has a unique nonnegative root ."/ .0/ which does not depend on the choice of the state j 2 X1 and has the following

318

5

Nonlinearly perturbed semi-Markov processes

asymptotic expansion: ."/ D .0/ C a1 " C    C ak "k C o."k /;

(5.4.13)

where the coefficients an , n D 1; : : : ; k do not depend on the choice of state .0/ j 2 X1 and are given by the recurrence formulas a1 D bjj Œ.0/ ; 1; 0= bjj Œ.0/ ; 0; 1 and, for n D 1; : : : ; k, an D  bjj Œ

.0/

; 0; 1

1

 n1 X .0/ bjj Œ ; n; 0 C bjj Œ.0/ ; n  q; 1aq qD1

C

n X

n X

bjj Œ

.0/

; n  q; m

mD2 qDm

X

q1 Y



n app =np Š

; (5.4.14)

n1 ;:::;nq1 2Dm;q pD1

where Dm;q , for every 2  m  q < 1, is the set of all nonnegative, integer solutions of the system n1 C    C nq1 D m;

n1 C    C .q  1/nq1 D q;

(5.4.15)

and the coefficients bjj Œ.0/ ; r; n, r D 0; : : : ; k  n, n D 0; : : : ; k, are given by the recurrence formulas (5.3.39) or (5.3.40) in Lemma 5.3.5 for every j 2 X1.0/ . ..0/ ;h/

(ii) If condition O5 condition ah > 0.

.0/ O6. ;h/

holds for some 1  h  k, then a1 ; : : : ; ah D 0. If

holds for some 1  h  k, then a1 ; : : : ; ah1 D 0 but

.r/

(iii) If condition B5 holds for some 1  r  k, then the following asymptotic relation holds for i; l ¤ 0: ."/

Pi f."/ .t ."/ / D l; 0 > t ."/ g expf..0/ C a1 " C    C ar1 "r1 /t ."/ g .0/ ! QQ il ..0/ /e r ar as " ! 0:

(5.4.16) .0/

Remark 5.4.3. In the asymptotic relation (5.4.16), the quantities Q l ..0/ / and .0/ .0/ .0/ QQ ..0/ / D 0  ..0/ /Q ..0/ / are given by formulas (4.6.27) and (4.6.26). il

il

l

Proof. The proof is based on a use of the corresponding Theorems 3.4.4 and 3.4.9 for perturbed regenerative processes or on the improvement of the corresponding asymptotic relations given in Theorems 4.6.3 and 4.6.7 for perturbed semi-Markov processes. The latter theorems yield, under conditions A2 , T1 , E2 , C12 , C13 , and N1 , the following asymptotic relation for i; l ¤ 0: ."/ g Pi f."/ .t ."/ / D l; ."/ .0/ 0 >t ! QQ il ..0/ / as " ! 0: ."/ ."/ expf t g

(5.4.17)

5.4 Exponential expansions for nonlinearly perturbed semi-Markov processes

319

The asymptotic relation (5.4.17) should be supplemented with the asymptotic expansion (5.4.13) for the root ."/ of the characteristic equation (5.4.2). This can be ."/ done by using Theorem 2.2.1, where the distribution function 0 Gjj .t / replaces the ."/ distribution function F .t / in the characteristic equation (2.1.4). Note that, accord.0/ ing to Lemma 4.5.6, the root ."/ does not depend on the choice of the state j 2 X1 . Lemmas 4.2.3, 4.5.6, and 4.5.8 imply that, under conditions A2 , T1 , E2 , C12 , C13 , ."/ and N1 , the distribution function 0 Gjj .t / satisfies conditions D11 and C2 . Lemmas .k/

5.3.5 and 5.3.6 also imply that the perturbation condition P3 holds under conditions .0/

;k/ . The mixed power-exponential moments of hitA2 , T1 , E2 , C12 , C13 , and P. 25 ."/ .0/ ting times, 0 jj Π; r, r D 0; : : : ; k, replace mixed moment generating functions .k/

 ."/ ..0/ ; r/, r D 0; : : : ; k, in condition P3 . The asymptotic expansion (2.2.2) takes in this case a form of the asymptotic expansion (5.4.13). This proves statement (i) of the theorem. Lemma 5.3.9 implies the statement (ii) of the theorem. By substituting the asymptotic expansion (5.4.13) in the asymptotic relation (5.4.17), we transform this relation to the following form: ."/

Pi f."/ .t ."/ / D l; 0 > t ."/ g .0/ ! QQ il ..0/ / as " ! 0: (5.4.18) .0/ k k ."/ expf. C a1 " C    C ak " C o." //t g .r/

If the balancing condition B5 holds for some 1  r  k, then (a) expfar "r t ."/ g ! expfar r g as " ! 0, and (b) expf.ar C1 "rC1 C    C ak "k C o."k //t ."/ g ! 1 as " ! 0. Using (a) and (b) we can transform relation (5.4.18) into the desirable form (5.4.16). This proves the statement (ii) of the theorem. Let us introduce the integer parameters h  1, k0  h, nQ D n.h/ Q D max.n W nh  Q k0 /, and k0  k1      knQ  1, and denote k.h/ D .n.h/; Q k0 ; : : : ; kn.h/ /. Q Theorem 5.4.4. Let conditions A2 , T1 , E2 , C12 , C13 , N1 , O5. hold. Then

.0/ ;h1/

. , and P27

.0/ ;k Q .h//

(i) For all " small enough there exists a unique nonnegative root ."/ of equation .0/ (5.4.2), it does not depend on the choice of the state j 2 X1 , and has the following asymptotic expansion: ."/ D .0/ C ah " C    C ak "k C o."k /;

(5.4.19)

where k D min.k0 ; k1 C h; : : : ; kn.h/ C hn.h//, Q the coefficients an , n D Q .0/ h; : : : ; k, do not depend on the choice of the state j 2 X1 , and are given by the

320

5

Nonlinearly perturbed semi-Markov processes

recurrence formulas ah D bjj Œ.0/ ; h; 0=bjj Œ.0/ ; 0; 1 and, for n D h; : : : ; k,  n1 X an D bjj Œ.0/ ; 0; 11 bjj Œ.0/ ; n; 0 C bjj Œ.0/ ; n  q; 1aq qDh

C

Œn= Xh

n X

bjj Œ

.0/

; n  q; m 

mD2 qDmh

X

q1 Y



n app =np Š

; (5.4.20)

n1 ;:::;nq1 2Dh;m;q pD1

where Dh;m;q , for every h  1 and 2  m  q < 1, is the set of all nonnegative, integer solutions of the system nh C    C nq1 D m;

hnh C    C .q  1/nq1 D q;

(5.4.21)

and the coefficients bjj Œ.0/ ; r; n, r D 0; : : : ; kn , n D 0; : : : ; n.h/, Q are given, .0/ for every j 2 X1 , by the recurrence formulas (5.3.39) or (5.3.40) in Lemma 5.3.5. ..0/ ;hQ /

(ii) If condition O5

..0/ ;hQ /

condition O6 ahQ > 0.

holds for some h  hQ  k, then ah ; : : : ; ahQ D 0. If holds for some h  hQ  k, then ah ; : : : ; ah1 D 0 but Q

(iii) If condition B.r/ 5 holds for some 1  r  k, then the following asymptotic relation holds for i; l ¤ 0: ."/

Pi f."/ .t ."/ / D l; 0 > t ."/ g expf..0/ C ah "h C    C ar1 "r1 /t ."/ g .0/ ! QQ il ..0/ /e r ar as " ! 0:

(5.4.22)

Proof. The proof uses the corresponding Theorems 3.4.5 and 3.4.31 for perturbed regenerative processes and the improvement of the corresponding asymptotic relations given in Theorems 4.6.1 and 4.6.2 for perturbed semi-Markov processes. ."/ Again, the distribution function 0 Gjj .t / plays the role of the distribution function F ."/ .t / in the characteristic equation (2.1.4). The only difference with the proof of Theorem 5.4.3 is that the corresponding asymptotic expansion for the characteristic root ."/ of the characteristic equation (5.4.2) takes another form with the coefficients a1 ; : : : ; a  h  1 equal zero. This can be done by using Theorem 2.2.2 instead of Theorem 2.2.1. Lemmas 5.1.6, 5.1.9, and 5.2.1 imply that, under conditions A2 , T1 , ..0/ ;kQ .h// .kQ .h// , the perturbation condition P4 holds. The asymptotic E2 , C12 , C13 , and P27 expansion (2.2.20) takes the form of the asymptotic expansion (5.4.19). This proves statement (i) of the theorem. The proof of statements (ii) and (iii) is analogous to those given for Theorem 5.4.3.

321

5.5 Asymptotics of quasi-stationary distributions

5.5 Asymptotics of quasi-stationary distributions for nonlinearly perturbed semi-Markov processes In this section, we give asymptotic expansions for stationary and quasi-stationary distributions of perturbed semi-Markov processes.

5.5.1 Cyclic representation for stationary distribution of ergodic semiMarkov process In this section we consider a model without absorption, where condition A4 holds, i.e., P ."/ (a) there exists "1 > 0 such that i¤0 pi0 D 0 for "  "1 . ."/

.0/

We also assume that condition T1 (a) holds, i.e., (b) pij ! pij as " ! 0 for i; j ¤ 0. Condition E2 is also assumed to hold. According to Lemma 4.2.2, conditions T1 (a), A4 , and E2 imply that (c) there exists "0 > 0 such that the corresponding sets of ."/ ."/ recurrent states X1 for the imbedded Markov chains n , n D 0; 1; : : :, are connected .0/ ."/ by the relation X1 X1 for every "  "0 . ."/ .0/ We assume that condition M.1/ 13 holds, i.e., (d) pij Œ1 ! pij Œ1 < 1 as " ! 0 for i; j ¤ 0. ."/ ."/ ."/ This condition implies that (e) there exists "2 > 0 such pij Œ1 D pij mij < 1 for i; j ¤ 0 and "  "2 . Let us now introduce the expectations of sojourn times, X ."/ X ."/ ."/ ."/ pij Œ1 D pij mij ; i ¤ 0: mi D j 2X

j 2X ."/

."/

."/

Note that, due to condition A4 , moments pi0 Œ1 D pi0 mi0 D 0, i ¤ 0 for "  "1 . ."/ .0/ Relations (b)–(e) imply that (f) mi ! mi as " ! 0 for i ¤ 0. Finally, we assume that condition I4 holds. Obviously, statement (e) implies that (g) m.0/ < 1; i ¤ 0. In this case, condition I4 is equivalent to the relation (h) P i .0/ > 0. .0/ m i i2X1 P Relations (f)–(h) imply that (i) there exists "3 > 0 such that (j) i2X .0/ m."/ i > 0 1 for "  "3 . .1/ If conditions A4 , T1 (a), E2 , M13 , and I4 hold, then the semi-Markov process ."/ .t /, t  0, is ergodic for every "  "4 , where "4 D min."0 ; "1 ; "2 ; "3 /. The semi-Markov process ."/ .t /, t  0, can be characterized as an ergodic process since, under the above conditions, the following relation holds for every "  "4 and i; l ¤ 0:   Z 1 t ."/ ."/ Pi lim D 1: (5.5.1) . .s/ D l/ ds D l t!1 t 0

322

5

Nonlinearly perturbed semi-Markov processes

."/

Here l , l ¤ 0, is a stationary distribution of the semi-Markov process ."/ .t /, ."/

t  0. This distribution is given by the following formula for j 2 X1 : ."/

l

D

Mjj."/l Mjj."/

;

l ¤ 0;

(5.5.2)

where ."/ Mij

D

."/ Ei j ;

."/ Mij l

Z D Ei

."/

j 0

.."/ .s/ D l/ ds;

i; j; l ¤ 0:

It is clear that the stationary distribution satisfies the relations (k) l."/  0, l ¤ 0, P and (l) l¤0 l."/ D 1. Also, (m) the stationary distribution does not depend on the choice of the state ."/ j 2 X1 used to calculate the stationary distribution by formula (5.5.2). Note also that (n) the limits in the relation (5.5.1) do not depend on the choice of the initial state i ¤ 0. A proof of the ergodic theorem, which uses the asymptotic relation (5.5.1), can be found, for example, in Silvestrov (1980a). Here we give a sketch of this proof. ."/ Choose and fix a recurrent state j 2 X1 . For this state, we will carry out a cyclic construction as described below. For every t  0, we have j."/ .t/

X

Z

."/

j l .n/ 

nD1 ."/

where (o1 ) j .n/ D

."/ ."/

j .n/

t 0

.."/ .s/ D l/ ds 

j."/ .t/C1

X

."/

j l .n/;

(5.5.3)

nD1

, n D 0; 1; : : : are the subsequent moments the semi."/

."/

Markov process ."/ .s/ hits the state j ; (o2 ) j .0/ D 0, j .n/ D min.k > ."/

."/

j .n  1/, k D j /, n D 1; 2; : : :, are subsequent moments at which the imbed."/

."/

."/

ded Markov chain n hits the state j ; (o3 ) j .t / D max.n W j .n/  t / is, for given t  0, the number of times the process ."/ .s/ hits the state j in the time ."/ ."/ ."/ interval Œ0; t ; (o4 ) j .n/ D j .n/  j .n  1/, n D 1; 2; : : :, are times between subsequent moments at which the process ."/ .s/ hits the state j ; and (o5 ) R j."/ .n/ ."/ .."/ .s/ D l/ ds, n D 1; 2; : : :, are times spent by the process

j l .n/ D ."/ j .n1/

."/ .s/ in the state l between subsequent moments the process hits the state j . ."/ By the definition, the random variables j .n/, n D 1; 2; : : :, are independent, ."/

."/

identically distributed at least for n  2, and Ej j .2/ D Mjj . Pi flimn!1 n

1

."/ j .n/

D

."/ Mjj g

D 1 for i ¤ 0, j 2

."/ X1 .

Thus, (p)

It easily follows from (p)

323

5.5 Asymptotics of quasi-stationary distributions ."/

."/

and the definition of the random variables j .t / that (q) Pi flim t!1 t 1 j .t / D ."/

."/

.Mjj /1 g D 1 for i ¤ 0, j 2 X1 . By the definition, the random variables ."/

j l .n/, n D 1; 2; : : :, are independent, identically distributed at least for n  2, P ."/ ."/ ."/ ."/ and Ej j l .2/ D Mjj l . Thus, (r) Pi flimn!1 n1 nrD1 j l .r/ D Mjj l g D 1 for Pj."/.t/ ."/ ."/

j l .n/ D i ¤ 0, j 2 X1 . Relations (q)–(r) imply that (s) Pi flim t!1 t 1 nD1 ."/

."/

."/

Mjj l .Mjj /1 g D 1 for i ¤ 0, j 2 X1 , and also that (t) Pi flim t!1 t 1  Pj."/ .t/C1 ."/ ."/ ."/ ."/

j l .n/ D Mjj l .Mjj /1 g D 1 for i ¤ 0, j 2 X1 . Relations (s), (t) and nD1 R1 ."/ ."/ (5.5.3) imply that (u) Pi flim t!1 t 1 0 .."/ .s/ D l/ ds D Mjj l .Mjj /1 g D 1 ."/

for i ¤ 0, j 2 X1 . ."/ Note also that claim (m) holds, since any recurrent state j 2 X1 can be chosen as to realise the cyclic construction described above. Thus, almost sure convergence of Rt ."/ ."/ the random variables 1t 0 .."/ .s/ D l/ ds in (5.5.1) to the quantity Mjj l .Mjj /1

can be proved for every j 2 X1."/ . Of course, this is possible only if these quantities ."/ take the same value for every j 2 X1 . Claim (n) follows from the above proof. As a matter of fact, the choice of the initial state i ¤ 0 influences the random variables introduced in (o1 )–(o5 ) only via the ."/ ."/ random variables j .1/ and j l .1/. These random variables have distributions dif."/

."/

ferent from the random variables j .2/ and j l .2/, respectively, if i ¤ j . However, this does not effect the corresponding asymptotic relations (p)–(u), since these random variables, normalised in an appropriate way, converge almost surely to zero. Relation (5.5.1) can be connected with the “individual” ergodic theorem given in Theorem 4.4.3. Note that the variables that enter this relation are bounded, more precisely, we have Z 1 t 0 .."/ .s/ D l/ ds  1; t  0; l ¤ 0: (5.5.4) t 0 Relations (5.5.1) and (5.5.4) imply, by the Lebesgue and Fubini theorems, that, for every "  "4 and i; l ¤ 0, Z Z 1 t 1 t ."/ Ei . .s/ D l/ ds D Pi f."/ .s/ D lg ds ! l."/ as t ! 1: (5.5.5) t 0 t 0 Theorem 4.4.3 also needs to assume the additional non-arithmetic condition N2 ."/ which guarantees the existence of "5 > 0 such that the distributions 0 Gjj .t /; j 2 ."/

X1 , are non-arithmetic for "  "5 . .1/ If conditions A4 , T1 (a), E2 , M13 , I4 , and N2 hold, Theorem 4.4.3 yields the following asymptotic relation for every "  min."4 ; "5 / and i; l ¤ 0: ."/

Pi f."/ .t / D lg ! l

as t ! 1:

(5.5.6)

324

5

Nonlinearly perturbed semi-Markov processes

This relation is obviously stronger than relation (5.5.5), since (5.5.6) implies (5.5.5) but not vice versa. In fact, Theorem 4.4.3 gives a stationary distribution in the form given in formula (4.4.28). However, the semi-Markov process ."/ .t /, t  0, is a regenerative process with times of returns to any fixed recurrent state considered as regeneration times. Theorem 4.4.3 is just a particular case of the “individual” ergodic Theorems 3.2.3 and 3.2.7 for ."/ regenerative processes. These theorems, applied to the case where a state j 2 X1 is taken for constructing regeneration cycles and a one-state set A D flg is chosen in the asymptotic relation (3.2.15), give the following formula for a stationary distribution of the ergodic semi-Markov process ."/ .t /, t  0: R1 ."/ ."/ 0 Pj f .s/ D l; j > sg ds ."/ ; l ¤ 0: (5.5.7) l D Mjj."/ Both forms given in (4.4.28) and (5.5.7) coincide, since they give limits of the same probabilities Pi f."/ .t / D lg. Also, the form of the stationary distribution given in formula (5.5.2) coincides with that given in (5.5.7), since the Ces`aro limit in (5.5.5) must coincide with the limit in (5.5.6). The same is implied by the following relation obtained with the use of the Fubini theorem: Z ."/ Z 1 j ."/ ."/ .."/ .s/ D l/ ds D Ej .."/ .s/ D l; j > s/ ds Mjj l D Ej 0 1

Z D Z D

."/

Ej .."/ .s/ D l; j

0 1 0

0

."/

Pj f."/ .s/ D l; j

> s/ ds

> sg ds:

(5.5.8)

It also useful to note that formula (5.5.7) is a variant of the corresponding formula (3.2.9) for regenerative processes, where the Lebesgue integration is used. But the ."/ function Pi f."/ .s/ D l; j > sg is directly Riemann integrable on Œ0; 1/. Indeed, this function has at most a countable set of discontinuity points, since the process ."/ .s/, s  0 has continuous from the right stepwise trajectories. Also, this function is non-negative and bounded from above by the Riemann integrable monotone function ."/ Pi fj > sg. Thus, the Lebesgue integration can be replaced with the Riemann integration in the integral in the numerator of the fraction in (5.5.7) and in the integrals that enter (5.5.8).

5.5.2 Asymptotic expansions for expectations of times spend in a given state before hitting a given state The cyclic representation formula (5.5.2) for the stationary distribution plays a key role in what follows.

5.5 Asymptotics of quasi-stationary distributions

325

.k/

We assume that the perturbation condition P18 (introduced in Subsection 5.1.1) holds for the transition probabilities of the imbedded Markov chains, .k/ ."/ .0/ : pij D pij C eij Œ1; 0" C    C eij Œk; 0"k C o."k / for i; j ¤ 0, where P18 jeij Œr; 0j < 1; r D 1; : : : ; k.

We also assume that the following perturbation condition holds for the expectation of transition times: .k/

."/

.0/

P28 : pij Œ1 D pij Œ1 C eij Œ1; 1" C    C eij Œk; 1"k C o."k / for i; j ¤ 0, where .0/

pij Œ1 < 1; i; j ¤ 0, and jeij Œr; 1j < 1, r D 1; : : : ; k, i; j ¤ 0. .k/

.k/

Obviously condition P28 is implied by the condition P18 and the following condition: 0

.0/ k k P28.k/ : m."/ ij D mij C vij Œ1; 1" C    C vij Œk; 1" C o." / for i; j ¤ 0, where

m.0/ ij < 1, i; j ¤ 0, and jvij Œr; 1j < 1, r D 1; : : : ; k, i; j ¤ 0.

Moreover, the following formula connect the corresponding asymptotic expansions, for i; j ¤ 0 and r D 0; : : : ; k, eij Œr; 1 D

r X

eij Œm; 0vij Œr  m; 1;

(5.5.9)

mD0 .0/ .0/ where eij Œ0; 1 D pij Œ1; eij Œ0; 0 D pij , and vij Œ0; 1 D m.0/ ij . .k/

.k/

Condition P18 obviously implies that condition T1 (a) holds and condition P28 .1/ implies condition M13 to hold. The following lemmas, which are direct corollaries of statements (i) and (ii) in Lemma 8.1.1, gives asymptotic expansions for the expectations m."/ i , i ¤ 0. .k/ .k/ and P28 hold. Then Lemma 5.5.1. Let conditions P18

(i) The following expansion holds for i ¤ 0: ."/

mi

.0/

D mi

C vi Œ1; 1" C    C vi Œk; 1"k C o."k /;

(5.5.10)

where the coefficients vi Œr; 1, r D 0; : : : ; k, are given for i ¤ 0 by the formulas vi Œ0; 1 D m.0/ i and, for r D 0; : : : ; k, vi Œr; 1 D

X j ¤0

eij Œr; 1:

(5.5.11)

326

5

Nonlinearly perturbed semi-Markov processes

0 .k/

.k/

(ii) If condition P28 holds instead of condition P28 , then the formulas (5.5.11) for the coefficients in the asymptotic expansion (5.5.10) can be rewritten in the following form: vi Œr; 1 D

X

eij Œr; 1 D

r XX

eij Œq; 0vij Œr  q; 1:

(5.5.12)

j ¤0 qD0

j ¤0

The asymptotic expansion (5.5.10) can be rewritten in a vector form. Let us introduce the following vectors for l ¤ 0 and r D 1; : : : ; k: 2

v."/

."/

m1 6 D 4 :::

3

3 v1 Œr; 1 7 6 :: vŒr; 1 D 4 5: : vN Œr; 1 2

7 5;

."/

mN

.k/

Then the perturbation condition P28 gives the asymptotic vector expansion v."/ D v.0/ C vŒ1; 1" C    C vŒk; 1"k C o."k /:

(5.5.13)

Note that, under condition A4 , Mjj."/ D 0 Mjj."/ , j 2 X1."/ . The asymptotic expansions for these expectations are given in Lemma 5.2.2. Note that it is necessary to set nQ D 1, kQ0 D kQ1 D k in this lemma in order to get asymptotic expansions of the order k for these expectations. Similar asymptotic expansions can be given for the expectations Mjj."/l . The following representation can be written for l ¤ 0: Z



."/

j 0

.."/ .s/ D l/ ds D

."/

j X

."/

n."/ .n1 D l/:

(5.5.14)

nD1 ."/

This representation immediately shows that the expectations Mij l , i ¤ 0, satisfy .0/

the following system of linear equations for every j 2 X1 and l ¤ 0: X ."/ ."/ Mij."/l D m."/ ı.i; l/ C pi i 0 Mi 0 j l ; i ¤ 0: i

(5.5.15)

i 0 ¤0;j .0/

These systems can be rewritten in a matrix form. For j 2 X1 , l ¤ 0, r D 1; : : : ; k, introduce the vectors 2 3 3 2 ."/ 2 ."/ M1j l m1 ı.1; l/ v1 Œr; 1ı.1; l/ 6 7 7 6 6 ."/ ."/ : : 6 7 : ::: : mjl D 4 5 ; vl Œr; 1 D 4 : : 5 ; vl D 4 vN Œr; 1ı.N; l/ m."/ ı.N; l/ M ."/ Nj l

N

and 3 7 5:

327

5.5 Asymptotics of quasi-stationary distributions .k/

Then the perturbation condition P28 implies that, for l ¤ 0, ."/

vl

.0/

D vl

C vl Œ1; 1" C    C vl Œk; 1"k C o."k /:

(5.5.16)

.0/

It is also convenient to set vi Œ0; 1 D mi , i ¤ 0, and vŒ0; 1 D v.0/ , and vl Œ0; 1 D .0/ vl , l ¤ 0. Then the system of linear equations (5.5.15) can be rewritten in the following form .0/ for every j 2 X1 and l ¤ 0: ."/ ."/ ."/ mjl : m."/ jl D vl C j P

(5.5.17)

These linear systems are systems of the hitting type, considered in Section 5.1, with the same coefficient matrix j P."/ . .1/ Conditions A4 , E2 , T1 (a), and M12 imply that for every "  "4 there exists a unique solution of the systems of linear equations (5.5.17), which has the following .0/ form for "  "4 and j 2 X1 and l ¤ 0: ."/ 1 ."/ m."/ jl D ŒI  j P  vl :

(5.5.18)

The following lemma is a corollary of Lemma 5.1.3 applied to the vectors of the .0/ power moments of hitting times, m."/ jl , for j 2 X1 and l ¤ 0. .k/

.k/

Lemma 5.5.2. Let conditions A4 , E2 , P18 , and P28 hold. Then ."/

.0/

(i) The vector mjl has the following asymptotic expansion for every j 2 X1 and l ¤ 0: ."/

.0/

mjl D mjl C cjlŒ1; 0" C    C cjl Œk; 0"k C o."k /;

(5.5.19)

where cjl Œr; 0, r D 1; : : : ; k, are finite vectors. (ii) The vector coefficients cjl Œr; 0 are given by the recurrence formulas cjlŒ0; 0 D .0/ .0/ mjl D j UŒ0vl Œ0; 1 D ŒI  j P.0/ 1 vl and, for r D 0; : : : ; k, cjl Œr; 0 D j UŒ0.vl Œr; 1 C

r X

j EŒq; 0cjl Œr

 q; 0/:

(5.5.20)

qD1

(iii) The vector coefficients cj Œr; 0 can also be calculated from the following formulas for r D 0; : : : ; k: cjlŒr; 0 D

r X pD0

j UŒp vl Œr

 p; 1:

(5.5.21)

328

5

Nonlinearly perturbed semi-Markov processes ."/

It is useful to note that the vectors mj by the formula ."/

mj

D

."/

.0/

and mjl , l ¤ 0, for j 2 X1 are connected X

."/

mjl :

(5.5.22)

l¤0 ."/

."/

The corresponding asymptotic expansions for the vectors mj D 0 mj are given in Lemma 5.2.2, where we must take nQ D 1, kQ0 D kQ1 D k. The coefficients in these expansions are given in formulas (5.2.19) and (5.2.20), where r D 1. Formula (5.5.22) implies that the corresponding coefficients in the asymptotic ex.0/ pansions for the vectors m."/ D 0 m."/ and m."/ j j jl , l ¤ 0, are connected for j 2 X1 and r D 0; : : : ; k by the formula X cjl Œr; 0: (5.5.23) 0 bj Œr; 1 D l¤0

5.5.3 Pivotal properties of asymptotic expansions for expectations of times spend in a given state before hitting a given state We will clarify pivotal properties of asymptotic expansions for the cyclic expectations ."/ .0/ Mjj l , l ¤ 0, j 2 X1 . .0/

For the vectors cjl Œr; 0, r D 0; : : : ; k, l ¤ 0, j 2 X1 , we denote 3 2 c1j l Œr; 0 7 6 :: cjlŒr; 0 D 4 5: : cNj l Œr; 0 Explicit expressions for components of these vectors can be derived from recurrence formulas (5.5.20) and (5.5.21), respectively. Consider the sets ( fl ¤ 0 W vl Œn; 1 D 0; n D 0; : : : ; r  1; vl Œr; 1 ¤ 0g for r D 1; : : : ; k, YQr D for r D k C 1. fl ¤ 0 W vl Œn; 1 D 0; n D 0; : : : ; kg By the definition, (a) some of the sets YQr , r D 0; : : : ; k C 1, could be empty, (b) S Q Q Yr 0 \ YQr 00 D ¿, r 0 ¤ r 00 , but (c) kC1 r D0 Yr D X0 . .0/ For j 2 X1 , define the sets [ [ .YQr 0 \ Zr 00 /; r D 0; : : : ; k; WQ kC1 D .YQr 0 \ Zr 00 /: WQ r D r 0 Cr 00 Dr

r 0 Cr 00 kC1

The case where condition E02 holds is simple. Indeed, in this case, Z0 D X1.0/ D X0 , and, therefore, Z1 ; : : : ; ZkC1 D ¿. Thus, for j 2 X1.0/ , we have WQ r D YQr ;

r D 0; : : : ; k C 1:

(5.5.24)

329

5.5 Asymptotics of quasi-stationary distributions

For an integer 0  h  k, introduce two conditions, l … WQ r , for r  h;

.l;h/ O7 :

and .l;h/ O8 : l … WQ r , for r < h, but l 2 WQ h . The following lemma gives a complete description of pivotal properties of asymp."/ .0/ totic expansions for the cyclic expectations Mjj l , l ¤ 0, j 2 X1 . .k/ .k/ , and P28 hold. Then Lemma 5.5.3. Let conditions A4 , E2 , P18 .l;h/

(i) If condition O7 holds for some l ¤ 0 and 0  h  k, then cjj l Œn; 0 D 0, .0/ n D 0; : : : ; h, j 2 X1 . .l;h/

(ii) If condition O8 holds for some l ¤ 0 and 0  h  k, then cjj l Œn; 0 D 0, .0/ .0/ n D 0; : : : ; h  1, j 2 X1 , but cjj l Œh; 0 > 0, j 2 X1 . .0/

."/

Proof. Recall that X1 X1 for "  "4 . Also, according to formula (5.5.18), the ."/ cyclic components Mjj l can be calculated by the following formula for every l ¤ 0, .0/

j 2 X1 : ."/

."/

."/

Mjj l D j uj l ml :

(5.5.25)

S Q Q Since kC1 r D0 Wr D X0 , there exists 0  r  k C 1 such that l 2 Wr . Let us first consider the case where r  k. Then (a) there exist 0  r 0 ; r 00  r such that r 0 C r 00 D r and l 2 YQr 0 \ Zr 00 . and In this case, the cyclic components Mjj."/l are products of two functions j uj."/ l which have, respectively, a pivotal .r 00 ; k/-expansion and a pivotal .r 0 ; k/-expanm."/ l

sion. According to claim (ii) in Lemma 8.1.1, (b) Mjj."/l is a pivotal .r 0 C r 00 ; .r 0 C k/ ^ .r 00 C k//-expansion. Obviously such an expansion can be considered as a pivotal .r; k/-expansion. Let us now consider the case where r D k C 1. In this case, either (c) there exist 0  r 0 ; r 00  k such that r 0 C r 00  k C 1 and l 2 YQr 0 \ Zr 00 , or (d) there exist 0  r 0 ; r 00  k C 1, max.r 0 ; r 00 /  k C 1 and l 2 YQr 0 \ Zr 00 . ."/ ."/ In the case (c), the cyclic components Mjj l are products of two functions j uj l ."/

and ml

which have, respectively, a pivotal .r 00 ; k/-expansion and a pivotal .r 0 ; k/."/

expansion if r 0 ; r 00  k. Using claim (ii) in Lemma 8.1.1, we see that Mjj l is a pivotal .r 0 C r 00 ; .r 0 C k/ ^ .r 00 C k//-expansion. Since r 0 C r 00  k C 1, this means that (e) ."/ Mjj l is o."k /. ."/

."/

In the case (d), at least one of the functions j uj l or ml is o."k / and, therefore, (f)

their product Mjj."/l is o."k /.

330

5

Nonlinearly perturbed semi-Markov processes

.l;h/ If condition O7 holds for some 0  h  k, then (g) l 2 WQ r for some h C 1  ."/ r  k or (h) l 2 WQ kC1 . In the case (g), Mjj l is, by (b), a pivotal .r; k/-expansion and, ."/

therefore, cjj l Œn; 0 D 0, n D 0; : : : ; h. In the case (h), Mjj l is o."k / by (e) or (f) and, again, cjj l Œn; 0 D 0, n D 0; : : : ; h. This proves the first statement of the lemma. If condition O.l;h/ holds for some 0  h  k, then Mjj."/l is, by (b), a pivotal 8 .h; k/-expansion. This proves the second statement of the lemma. Remark 5.5.1. Note that condition I4 is not required either in Lemma 5.2.2 or in Lemma 5.5.2. However, this condition is necessary in ergodic theorems. If condition I4 holds, then there exists at least one state l 2 WQ 0 X1.0/ . The corresponding coefficients are cjj l Œ0; 0 > 0, j 2 X1.0/ .

5.5.4 Asymptotic expansions for stationary distributions of perturbed ergodic semi-Markov processes Now we are in a position to get asymptotic expansions for stationary distributions of perturbed ergodic semi-Markov processes. We will use the asymptotic expansions for ."/ ."/ ."/ .0/ the cyclic expectations Mjj D 0 Mjj and Mjj l , l ¤ 0, for j 2 X1 , given in Lemmas 5.2.2 and 5.5.2, and the cyclic representation for stationary distribution given in formula (5.5.2). .k/

.k/

Theorem 5.5.1. Let conditions A4 , E2 , I4 , P18 , and P28 hold. Then ."/

(i) The stationary probabilities l

.0/

."/

."/

D Mjj l =Mjj , l ¤ 0, which do not depend on

the choice of the state j 2 X1 , have the following asymptotic expansion for .0/ every l ¤ 0 and j 2 X1 : .0/

."/ l

D

Mjj l C cjj l Œ1; 0" C    C cjj l Œk; 0"k C o."k / .0/

Mjj C 0 bjj Œ1; 1" C    C 0 bjj Œk; 1"k C o."k / .0/

D l

C fl Œ1" C    C fl Œk"k C o."k /;

(5.5.26)

where the coefficients fl Œr are given by the recurrence formulas fl Œ0 D l.0/ D

Mjj.0/l =Mjj.0/ D cjj l Œ0; 0=0 bjj Œ0; 1 and, for n D 0; : : : ; k, fl Œn D .cjj l Œn; 0 

n1 X

0 bjj Œn

 q; 1fl Œq/=0 bjj Œ0; 1:

(5.5.27)

qD0 .l;h/

(ii) If condition O7 holds for given l ¤ 0 and 0  h  k, then cjj l Œn; 0 D 0, .0/ n D 0; : : : ; h, j 2 X1 , and fl Œn D 0, n D 0; : : : ; h.

331

5.5 Asymptotics of quasi-stationary distributions .l;h/

(iii) If condition O8 holds for given l ¤ 0 and 0  h  k, then cjj l Œn; 0 D 0, .0/ .0/ n D 0; : : : ; h  1, j 2 X1 , but cjj l Œh; 0 > 0, j 2 X1 , and fl Œn D 0, n D 0; : : : ; h  1, but fl Œh > 0. ."/

Proof. Lemmas 5.2.2 and 5.5.2 yield asymptotic expansions for the functions Mjj l ."/

and Mjj , which permits to imply statement (iii) of Lemma 8.1.1 to the quotient of .0/

these functions for every l ¤ 0 and j 2 X1 . This yields asymptotic expansions for ."/ the stationary probabilities l , i ¤ 0, given in Theorem 5.5.1. The pivotal properties described in statements (ii) and (iii) directly follow from Lemma 5.5.3 and statement (iv) in Lemma 8.1.1.

5.5.5 Cyclic representation for quasi-stationary distribution of semi-Markov process In this section we consider a model in which the absorption is possible both for perturbed and for the corresponding limit semi-Markov processes. ."/ .0/ We assume that condition T1 holds, i.e., (a) pij ! pij as " ! 0 for i ¤ 0, ."/

.0/

j 2 X, and (b) Qij ./ ) Qij ./ as " ! 0 for i ¤ 0, j 2 X. We also assume that condition E2 holds, i.e., the set of recurrent-without-absorption states is not empty, and that condition A2 is satisfied, i.e., 0 is an absorption state. According to Lemma 4.2.2, relation (a) and conditions A2 , T1 (a), and E2 imply that (c) there exists "00 > 0 such that the corresponding sets of recurrent states, X1."/ , for the imbedded Markov chains ."/ n , n D 0; 1; : : :, are connected by the relation X1.0/ X1."/ for every "  "00 . ."/ We assume that condition C12 is satisfied, i.e., (d) lim"!0 pij .ı/ < 1; i ¤ 0, j 2 X for some ı > 0. ."/ Obviously condition C12 implies that (e1 ) there exists "01 > 0 such that pij .ı/ < 1 ."/

for i ¤ 0, j 2 X and "  "01 . As was shown in Subsection 5.3.5, (e2 ) pij Œˇ; n  ."/

."/

cn .ı; ˇ/pij .ˇ/, n D 0; 1; : : :, for any ˇ < ı. Thus, (e3 ) pij Œˇ; n < 1 for "  "01 , ."/ Œ; n is a derivative (in the variable i ¤ 0; j 2 X. Note also that (e4 ) the function pij

."/ Œ; 0 for   ˇ, "  "01 , n D 1; 2; : : :, i ¤ 0, j 2 X, ) of order n of the function pij For i ¤ 0, n D 0; 1; : : :, introduce the mixed power-exponential moment generating functions for sojourn times by X ."/ X ."/ ."/ ."/ ./ D p ./ D pij ij ./;  2 R1 ; i ij j 2X

and

."/ i Œ; n

D

X j 2X

j 2X

."/

pij Œ; n D

X j 2X

."/

pij

."/ ij Œ; n;

 2 R1 :

332

5

Nonlinearly perturbed semi-Markov processes ."/

."/

Note that, by the definition, i Œ; 0 D i ./, i ¤ 0. ."/ Relations (a) and (e3 ) imply that (f1 ) pi Œˇ; n < 1 for "  "01 , n D 0; 1; : : :, ."/ i ¤ 0. Note also that (f2 ) for every n D 1; 2; : : : and i ¤ 0, the function pi Œ; n is ."/ a derivative (in the variable ) of order n of the function pi Œ; 0 for   ˇ, "  "01 , i ¤ 0. Assume that condition C13 holds, i.e., (g) there exist i 2 X1.0/ and ˇi 2 .0; ı for ı .0/ .0/ defined in condition C12 such that 0 i.0/ i .ˇi / 2 .1; 1/ and 0 ki .ˇi / < 1, k 2 X2 . As follows from Lemmas 4.5.3 and 4.5.6, conditions A2 , T1 , E2 , C12 , and C13 ."/ imply that (h1 ) there exist 0 < ˇ < ı and "02 > 0 such that 0 ij .ˇ/ < 1 for "  "02 , ."/

i ¤ 0, j 2 X1 . As follows from Lemmas 4.5.3 and 4.5.6, conditions A2 , T1 , E2 , C12 , and C13 also ."/ ./ D 1 for j 2 X1."/ imply that (i1 ) there exists a unique root ."/ of the equation 0 jj and "  "02 , (i2 ) the root ."/ does not depend on the choice of j 2 X1."/ , (i3 ) the root ."/ satisfies the inequality ."/ < ˇ for "  "02 , (i4 ) ."/ ! .0/ as " ! 0. Choose an arbitrary .0/ < ˇ 0 < ˇ. ."/ ."/ As was shown in Subsection 5.3.5, (j1 ) 0 ij Œˇ; n  cn .ˇ; ˇ 0 / 0 ij .ˇ/, n D ."/

."/

0; 1; : : : . Thus, (j2 ) 0 ij Œˇ 0 ; n < 1 for "  "02 , n D 0; 1; : : :, i ¤ 0, j 2 X1 . Note ."/

also that (j3 ) the function 0 ij Œ; n is a derivative (in the variable ) of order n of the ."/

function 0 ij Œ; 0 for   ˇ, "  "02 , n D 1; 2; : : :, i ¤ 0. We will use the following notations introduced in Subsection 4.4.1 for the moment generating functions,  2 R1 , i; j ¤ 0: Z 1 ."/ ."/ ."/ ."/ .j ^0 / ij ./ D Ei e D e s Gij .ds/: 0

Also, consider the corresponding mixed power-exponential generating functions for  2 R1 , i; j; l ¤ 0 and n D 0; 1; : : :, Z 1 ."/ ."/ s n e s Gij .ds/: ij Œ; n D 0

."/ ij Œ; 0

."/ ij ./,

D i; j ¤ 0. By the definition, As follows from Lemmas 4.5.4 and 4.5.7, conditions A2 , T1 , E2 , C12 , and C13 imply that (k1 ) there exists "03 > 0 such that ij."/ .ˇ/ < 1 for "  "03 and i ¤ 0, j 2 X1."/ .

Similarly to inequalities (j2 ), it can be proved that (k2 ) ij."/ Œˇ 0 ; n  cn0 .ˇ; ˇ 0 /ij."/ .ˇ 0 /,

n D 0; 1; : : : . Thus, (k3 ) ij."/ Œˇ 0 ; n < 1 for "  "03 , n D 0; 1; : : :, i ¤ 0, j 2 X1."/ . Note also that (k4 ) the function ij."/ Œ; n is the derivative (in the variable ) of order

n of the function ij."/ Œ; 0 for   ˇ, "  "03 , n D 1; 2; : : :, i ¤ 0. Let us introduce the following moment generating functions for  2 R1 , i; j; l ¤ 0: Z 1 ."/ ."/ ."/ !ij l ./ D e s Pi f."/ .s/ D l; j ^ 0 > sg ds: 0

333

5.5 Asymptotics of quasi-stationary distributions

Consider the following mixed power-exponential moment generating functions for  2 R1 , i; j; l ¤ 0 and n D 0; 1; : : :: Z 1 ."/ ."/ ."/ !ij l Œ; n D s n e s Pi f."/ .s/ D l; j ^ 0 > sg ds: 0

."/

."/

By the definition, !ij l Œ; 0 D !ij l ./, i; j; l ¤ 0. For  2 R1 , i; j ¤ 0, consider the moment generating functions Z 1 ."/ ."/ ."/ !ij ./ D e s Pi fj ^ 0 > sg ds; 0

Also consider the following mixed power-exponential moment generating functions for  2 R1 , i; j; l ¤ 0 and n D 0; 1; : : :: Z 1 ."/ ."/ ."/ !ij Œ; n D s n e s Pi fj ^ 0 > sg ds: 0

."/ ."/ By the definition, !ij Œ; 0 D !ij ./, i; j; ¤ It is clear that, for  2 R1 and i; j ¤ 0, ."/

!ij ./ D

X

0.

."/

(5.5.28)

!ij l ./:

l¤0

The following formula plays a key role in what follows: Z 1 ."/ ."/ ."/ !ij ./ D e s Pi fj ^ 0 > sg ds 0

Z

D Ei Z D Ei

1 0

."/

."/

e s .j ^ 0 > s/ds

."/

(

."/

j ^0

0

s

e ds D

."/

.ij Œ; 0  1/=; ."/ ij Œ0; 1;

if  > 0,

(5.5.29)

if  D 0.

We have the following estimate for i; j ¤ 0 and n D 0; 1; : : :: Z 1 ."/ 0 0 0 ."/ e ˇs Pi fj."/ ^ ."/ !ij Œˇ n  cn 0 > sg ds D cn !ij .ˇ/;

(5.5.30)

0

where cn0 D cn .ˇ; ˇ 0 / D sups0 s n e .ˇ ˇ

0 /s

< 1. ."/

It follows from (k1 ) and relations (5.5.28), (5.5.29), (5.5.30) that (l1 ) !ij l Œˇ; n 

."/

."/

!ij Œˇ; n < 1 for "  "03 , n D 0; 1; : : :, and i; l ¤ 0, j 2 X1 . Also note ."/

."/

that (k2 ) the functions !ij l Œ; n and !ij Œ; n are derivatives (in the variable ) of ."/

."/

order n, respectively, of the functions !ij Œ; 0 and !ij Œ; 0 for   ˇ 0 , "  "03 , ."/

n D 1; 2; : : :, i ¤ 0, j 2 X1 .

334

5

Nonlinearly perturbed semi-Markov processes

Relation (i4 ) also implies that (l) there exists "04 > 0 such that ."/ < ˇ 0 for "  "04 . We assume that condition I4 holds. Together with conditions T1 and C12 , condition I4 implies that (m1 ) there exists "05 > 0 such that maxi2X ."/ i."/ ./ > 1 for 0 <   1

ˇ 0 , "  "05 . Also, conditions A2 , T1 , E2 , C12 , C13 and I4 imply that (n1 ) there exists "06 > 0 ."/ such that jj ./ > 1 for 0 <   ˇ 0 , "  "06 , j 2 X1."/ . Finally, we require the non-arithmetic condition N2 , which implies (o1 ) existence ."/ ."/ of "08 > 0 such that the distributions 0 Gjj .t /; j 2 X1 , are non-arithmetic for "  "06 . According to Theorem 4.6.8, if conditions A2 , T1 , E2 , C12 , C13 , I4 and N2 hold, then (p1 ) the semi-Markov process ."/ .t /, t  0, is quasi-ergodic for every "  min."07 ; "08 /, where "07 D min."00 ; "01 ; "02 ; "03 ; "04 ; "05 ; "06 /, in the sense that the following asymptotic quasi-ergodic relation holds for i; l ¤ 0: e

."/ t

."/ ."/ Pi f."/ .t ."/ / D l; 0 > t g ! QQ il .."/ / as t ! 1;

(5.5.31)

."/ where the limit quantities QQ il .."/ / are given by formulas (4.6.3), (4.6.26) and (4.6.27). We prefer however to use an alternative cyclic representation formula for the lim."/ its QQ il .."/ /. As was mentioned above, the semi-Markov process ."/ .t /, t  0, is a regenerative process with the times of returns to any fixed recurrent state considered as regeneration times. Theorem 4.6.8 is just a particular case of the “individual” quasi-ergodic Theorems 3.3.3 and 3.3.11 for regenerative processes. These theorems, ."/ applied to the case where a state j 2 X1 is chosen to construct regeneration cycles and a one-state set A D flg is chosen in the asymptotic relation (3.3.79), give the ."/ ."/ following formula for the limits QQ il .."/ / for i; l ¤ 0 and j 2 X1 :

.."/ /; QQ il."/ .."/ / D 0 ij."/ .."/ /Qj."/ l

(5.5.32)

where R1

."/

.."/ / Qj."/ l

D

!jj l .."/ / ."/ ."/ 0 jj . /

D

0

e

."/ s

."/

."/

Pi f."/ .s/ D l; j ^ 0 > sg ds : (5.5.33) R1 ."/ s G ."/ .ds/ 0 jj 0 se

Both forms of the quasi-stationary limits QQ il."/ .."/ / given in formulas (5.5.32) and (5.5.33) and in formulas (4.6.3), (4.6.26), and (4.6.27) coincide, since they give limits ."/ ."/ of the same quantities e  t Pi f."/ .t / D l; 0 > t g. Formulas (5.5.32) and (5.5.33) are variants of the corresponding formulas for regenerative processes (3.3.78) and (3.3.6), respectively. These formulas use the Lebesgue ."/ integration. But the function e  s Pi f."/ .s/ D l; j."/ ^ ."/ 0 > sg is directly Riemann integrable on Œ0; 1/. Indeed, it has at most a countable set of discontinuity points, since the process ."/ .s/, s  0 has continuous from the right stepwise trajectories. Also this function is non-negative and bounded from above by the directly

335

5.5 Asymptotics of quasi-stationary distributions ."/

."/

."/

Riemann integrable function e  s Pi fj ^0 > sg. Thus, the Lebesgue integration can be replaced in the numerator of the fraction in (5.5.33) with Riemann integration. ."/ Note also that the limits QQ .."/ / do not depend on the choice of the state j 2 il

."/

."/

X1 , because we can choose any recurrent state j 2 X1 for constructing a corre."/ sponding regenerative process. Thus, convergence of the quantities e  t Pi f."/ .t / D lg in (5.5.31) to the quantity QQ il."/ .."/ / can be proved for every j 2 X1."/ . Obviously, this is possible only if these quantities take the same value for every j 2 X1."/ .

A dependence on the initial state i ¤ 0 is possible in the quasi-stationary case for ."/ > 0. This dependence disappears in the pseudo-stationary case, where ."/ D 0. In this ."/ ."/ ."/ ."/ case, 0 ij .0/ D 1, i ¤ 0, j 2 X1 and Qj l .0/ D l , l ¤ 0, coincide with the ."/

corresponding stationary probabilities given in formula (5.5.7) for every j 2 X1 . Finally, it is useful to note that formulas (5.5.32) and (5.5.33) define the corresponding quasi-stationary quantities QQ il."/ .."/ / for "  "07 without the assumption that the non-arithmetic condition N2 holds. ."/ ."/ It can be proved that QQ il .."/ / do not depend on the choice of the state j 2 X1 by using the same method as in the proof of Lemma 4.6.1. Let us prove it for the process .0/ .t /, t  0. The same method can be used for the process ."/ .t /, t  0, corresponding to any other value of the parameter ". Thus, we assume that conditions A2 , E2 , C12 , C13 , and I4 hold for this process. ."/ Let us choose and fix some state i 2 X1 and perturb the transition probabilities .0/ ."/ .0/ Qij .t /; j 2 X, by replacing them with the convolutions Qij .t / D Q."/ .t /Qij .t /, j 2 X, where Q."/ .t / D .t ="/ ^ 1, t  0, is a uniform distribution function on the interval Œ0; ". Fix some "0 > 0 and, for 0  "  "0 , consider the family of per."/ turbed semi-Markov processes ."/ .t /, t  0, with the transition probabilities Qij .t /. Obviously, conditions T1 and N2 are satisfied by the perturbed semi-Markov processes ."/ .t /, t  0. Thus there exists "00 > 0 such that the asymptotic relation (5.5.31) takes ."/ place for these processes for "  min."0 ; "00 /. The corresponding limits QQ il .."/ / do ."/

."/

not depend on the choice of the state j 2 X1 . Here X1 is the set of recurrent states for the process ."/ .t /, t  0, introduced above. Note that this set is the same for all 0  "  "0 , since all processes ."/ .t /, t  0, have the same transition matrix of imbedded Markov chains. By making " ! 0 and performing the corresponding limit transition using Lemma 4.6.2, we can get the same formula for the process .0/ .t / as .0/ well as to prove that the quantities QQ il ..0/ / do not depend on the choice of the state .0/

j 2 X1 . ."/

."/

."/

Considering the quantities Qj l .."/ /, l ¤ 0, j 2 X1 , we see that jj .."/ / D 1, j 2 X1."/ , and, therefore, QQj."/ .."/ / D Qj."/ .."/ /, l ¤ 0, j 2 X1."/ . l l

336

5

Nonlinearly perturbed semi-Markov processes ."/

.0/

Thus, it is possible that Qj l .."/ /, l ¤ 0, depend on the choice of j 2 X1

in the

."/

quasi-stationary case where > 0. This dependence disappears in the pseudo-stationary case where ."/ D 0. In this ."/ ."/ case, Qj l .0/ D l , l ¤ 0, coincide with the corresponding stationary probabilities ."/

given in formula (5.5.7) for every j 2 X1 . According to Theorem 4.6.5, if conditions A2 , T1 , E2 , C12 , C13 , I4 and N2 hold, then the following asymptotic quasi-ergodic relation also holds for every "  min."07 ; "08 / and i; l ¤ 0: ."/ ."/ Pi f."/ .t / D l=."/ 0 > t g ! l . / as t ! 1;

(5.5.34)

where l."/ .."/ / D P

."/ QQ il .."/ /

QQ ."/ .."/ / r ¤0  ir

;

l ¤ 0:

(5.5.35)

Formula (5.5.34) defines a quasi-stationary distribution for the semi-Markov pro."/ cess ."/ .t /, t  0. The quasi-stationary distribution l .."/ /, l ¤ 0, is defined via ."/ the quantities QQ .."/ / that are dependent on the choice of the initial state i ¤ 0. il

."/

However, Lemma 4.6.1 shows that the quasi-stationary probabilities l .."/ /, l ¤ 0, do not depend on the choice of the initial state i ¤ 0. Substituting the expressions for QQ il."/ .."/ /, i; l ¤ 0, given in formulas (5.5.32) ."/

."/

and (5.5.33) into formula (5.5.35), canceling the factors 0 ij .."/ / and 0 jj .."/ / in the numerator and the denominator we obtain the following expressions for the quasi."/ stationary probabilities for i ¤ 0, j 2 X1 : ."/

l .."/ / D P

."/ QQj l .."/ /

QQ ."/ .."/ / r ¤0  jr

."/

DP

."/ Qjr .."/ / r¤0 

."/

DP

!jj l .."/ / ."/ ."/ r ¤0 !jjr . /

Qj l .."/ /

."/

D

!jj l .."/ / ."/

!jj .."/ /

;

l ¤ 0:

(5.5.36)

As follows from the remarks above, these probabilities do not depend on the choice ."/ of the initial state i ¤ 0 or the state j 2 X1 .

5.5.6 Perturbation conditions used in asymptotic expansions for quasistationary distributions ."/ The cyclic representation formula (5.5.32) for the quasi-stationary limits QQ il .."/ / ."/

and formula (5.5.34) for the quasi-stationary probabilities l .."/ / will play a key role in what follows.

337

5.5 Asymptotics of quasi-stationary distributions

We assume that conditions A2 , T1 , E2 , C12 , and C13 hold. .k/ (introduced in We will also assume that the following perturbation condition P18 Subsection 5.1.1) for the transition probabilities of imbedded Markov chains holds: .k/

."/

.0/

P18 : pij D pij C eij Œ1; 0" C    C eij Œk; 0"k C o."k / for i; j ¤ 0, where jeij Œr; 0j < 1, r D 1; : : : ; k. Note that this condition involves an asymptotic perturbation expansion for transition probabilities up to the degree k. Also, this condition automatically yields an ."/ asymptotic expansion for the local absorption probabilities pi0 , i ¤ 0, given in the asymptotic relation (5.1.37). Consider the following perturbation condition for mixed power-exponential moment of transition times for   ˇ0 : .;k/ ."/ .0/ : pij Œ; n D pij Œ; n C eij Œ; 1; n" C    C eij Œ; k; n"kn C o."kn / for P29 n D 0; : : : ; k, i ¤ 0, j 2 X, where jeij Œ; r; nj < 1, r D 1; : : : ; k  n, n D 0; : : : ; k, i ¤ 0, j 2 X. .;k/

.;k/

This condition resembles condition P25 . However, condition P29 includes also ."/ the asymptotic expansions for mixed power-exponential moments pi0 Œ; n, n D .;k/ 0; : : : ; k, i ¤ 0, while condition P25 does not. This is because of the moments ."/ Œ; n penetrate formulas for quasi-stationary distributions. pi0 ."/ Recall the formulas that connect the mixed power-exponential moments pij Œ; n and

."/ ij Œ; n,

for every n D 0; 1; : : :, i ¤ 0, j 2 X, and  2 R1 , ."/ ."/ Œ; n D pij pij

."/ ij Œ; n:

.;k/ .k/ is implied by condition P18 and the These formulas imply that condition P29 following condition: 0 .;k/

P29

:

."/ ij Œ; n

.0/

D ij Œ; n C vij Œ; 1; n" C    C vij Œ; k; n"kn C o."kn / for n D 0; : : : ; k, i ¤ 0, j 2 X, where jvij Œ; r; nj < 1, r D 1; : : : ; k  n, n D 0; : : : ; k, i ¤ 0, j 2 X. 0

.k/ Moreover, the corresponding asymptotic expansions in conditions P18 and P29.;k/ .;k/ are connected with the asymptotic expansion in condition P29 by the following formulas for i ¤ 0, j 2 X and r D 0; : : : ; k  n, n D 0; : : : ; k:

eij Œ; r; n D

r X

eij Œr; 0vij Œ; r  m; n:

(5.5.37)

mD0

Consider the mixed power-exponential moments of sojourn times for  2 R1 , n D 0; 1; : : :, and i ¤ 0, X ."/ X ."/ ."/ ."/ Œ; n D p Œ; n D pij ij Œ; n: i ij i2X

i2X

338

5

Nonlinearly perturbed semi-Markov processes

The following lemma, which is the direct corollary of statements (i) and (ii) in Lemma 8.1.1, gives asymptotic expansions for the expectations i."/ Œ; n, i ¤ 0. .k/

.;k/

Lemma 5.5.4. Let condition P18 , together with condition P29 hold. Then

for some   ˇ0 ,

(i) The following asymptotic expansion takes place for n D 0; 1; : : : ; k and i ¤ 0: ."/ i Œ; n

D

.0/ i Œ; n C

vi Œ; 1; n" C   

C vi Œ; k  n; n"kn C o."kn /;

(5.5.38)

where the coefficients vi Œ; r; n, r D 0; : : : ; k  n, n D 0; : : : ; k, i ¤ 0, are .0/ given by the formulas vi Œ; 0; n D i Œ; n, n D 0; : : : ; k, i ¤ 0, and, for r D 0; : : : ; k  n, n D 0; : : : ; k, i ¤ 0, X eij Œ; r; n: (5.5.39) vi Œ; r; n D j 2X 0

.;k/ (ii) If condition P29.;k/ holds instead of condition P29 , then the formulas (5.5.39) for the coefficients in the asymptotic expansion (5.5.38) can be rewritten in the following form:

vi Œ; r; n D

X

eij Œ; r; n D

r XX

eij Œq; 0vij Œ; r  q; n:

(5.5.40)

j 2X qD0

j 2X

The asymptotic expansions (5.5.38) can be rewritten in a vector form. Let us introduce the following vectors for r D 0; : : : ; k C 1  n and n D 0; : : : ; k C 1: 3 3 2 ."/ 2 v1 Œ; r; n 1 Œ; n 7 7 6 6 :: :: § ."/ Œ; n D 4 vŒ; r; n D 4 5: 5; : : ."/ vN Œ; r; n Œ; n N

The asymptotic expansions (5.5.38) can be rewritten in the following vector form for r D 0; : : : ; k  n and n D 0; : : : ; k: § ."/ Œ; n D § .0/ Œ; n C vŒ; 1; n" C    C vŒ; k  n; n"kn C o."kn /: (5.5.41) For "  "06 and i ¤ 0, define the functions ( 1 . ."/ Œ; 0  1/ for  ¤ 0;   ˇ 0 , ."/ 'i Œ; 0 D  i."/ for  D 0, i Œ; 1 ( R1 ."/ 1 s .e  1/Fi .ds/ for  ¤ 0;   ˇ 0 , D R01  ."/ for  D 0. 0 sFi .ds/

(5.5.42)

5.5 Asymptotics of quasi-stationary distributions

339

."/

As was pointed out in Subsection 5.5.4, the functions i Œ; 0, i ¤ 0, have derivatives of any order n D 1; 2; : : : for "  "07 and   ˇ 0 . This implies that the functions ."/ 'i Œ; 0, i ¤ 0, have derivatives of any order n D 1; 2; : : : for "  "07 and   ˇ0 . We can obtain a relationship between these derivatives for "  "07 by differentiat."/ ."/ ing the relation (a) i Œ; 0 D 'i Œ; 0 C 1;   ˇ0 , i ¤ 0, that follows from formula (5.5.42). Differentiating (a) yields the following relation: (b) i."/ Œ; n D 'i."/ Œ; n C n'i."/ Œ; n  1, n D 1; 2; : : :,   ˇ 0 , i ¤ 0. In particular, (b) yields the following relation for  D 0: (c) i."/ Œ0; n D n'i."/ Œ0; n  1, n D 1; 2; : : :, i ¤ 0. Relations (a)–(c) yield the following recurrence relations for derivatives of the func."/ tions 'i Œ; n for n D 1; 2; : : :, "  "07 and   ˇ0 : ( ."/ 'i Œ; n

D

.

."/ ."/ i Œ; n  n'i Œ; n  ."/ i Œ0; n C 1=.n C 1/

1/=

for  ¤ 0;   ˇ 0 , for  D 0.

(5.5.43)

Lemmas 5.5.4 and 8.1.1 and formulas (5.5.42) and (5.5.43) easily yield asymptotic ."/ expansions for the functions 'i Œ; n, n D 0; 1; : : :, i ¤ 0 for the cases  ¤ 0 and  D 0. .k/

.;k/

Lemma 5.5.5. Let condition P18 hold condition P29 for some   ˇ0 ,  ¤ 0. Then the following asymptotic expansion takes place for n D 0; : : : ; k and i ¤ 0: ."/

.0/

'i Œ; n D 'i Œ; n C wi Œ; 1; n" C    C wi Œ; k  n; n"kn C o."kn /;

(5.5.44)

where the coefficients wi Œ; r; n, r D 0; : : : ; k  n, n D 0; : : : ; k are given for every i ¤ 0 by the formulas wi Œ; r; n ( 1 .vi Œ; r; 0  ı.r; 0// for r D 0; : : : ; k; n D 0; (5.5.45) D 1  .vi Œ; r; n  nwi Œ; r; n  1/ for r D 0; : : : ; k  n; n D 1; : : : ; k: .k/

.0;kC1/

hold. Then the following asymptotic Lemma 5.5.6. Let conditions P18 and P29 expansion takes place for n D 0; : : : ; k and i ¤ 0: 'i."/ Œ0; n D 'i.0/ Œ0; n C wi Œ0; 1; n" C    C wi Œ0; k  n; n"kn C o."kn /;

(5.5.46)

where the coefficients wi Œ0; r; n, r D 0; : : : ; k  n, n D 0; : : : ; k, are given for every

340

5

Nonlinearly perturbed semi-Markov processes

i ¤ 0 by the formulas ( for r D 0; : : : ; k; n D 0; vi Œ0; r; 1 wi Œ; r; n D 1 nC1 vi Œ0; r; n C 1 for r D 0; : : : ; k  n; n D 1; : : : ; k:

(5.5.47)

."/

Note that the asymptotic expansions of the order k for functions 'i Œ; n require .;k/ .0;kC1/ condition P29 , if  ¤ 0, and condition P29 , if  D 0. ."/ The asymptotic expansions (5.5.46) and (5.5.44) for functions 'i Œ; n can be rewritten in a vector form. For r D 0; : : : ; k  n and n D 0; : : : ; k, consider the vectors 3 3 2 ."/ 2 '1 Œ; n w1 Œ; r; n 7 7 6 6 :: :: ®."/ Œ; n D 4 wŒ; r; n D 4 5; 5: : : ."/ wN Œ; r; n ' Œ; n N

The asymptotic expansion (5.5.46) can be rewritten in the following vector form for n D 0; : : : ; k: ®."/ Œ; n D ®.0/ Œ; n C wŒ; 1; n" C    C wŒ; k  n; n"kn C o."kn /:

(5.5.48)

For r D 0; : : : ; k  n and n D 0; : : : ; k, l ¤ 0, introduce the vectors 3 3 2 ."/ 2 '1 Œ; nı.1; l/ w1 Œ; r; nı.1; l/ 7 7 6 6 :: :: ®."/ wl Œ; r; n D 4 5: 5; : : l Œ; n D 4 ."/ wN Œ; r; nı.N; l/ ' Œ; nı.N; l/ N

The asymptotic expansion (5.5.48) also gives the following asymptotic expansions for n D 0; : : : ; k, and l ¤ 0: ."/

.0/

®l Œ; n D ®l Œ; n C wl Œ; 1; n" C    C wl Œ; k  n; n"kn C o."kn /:

(5.5.49)

Also, it is convenient to define the vectors w.0/ Œ; 0; n D ®.0/ Œ; n and .0/ .0/ wl Œ; 0; n D ®l Œ; n for l ¤ 0 and n D 0; : : : ; k.

5.5.7 Asymptotic expansions for mixed power-exponential moments for the time spend in a given state before hitting a given state or an absorption Note that, under conditions A2 , T1 , E2 , C12 , and C13 , asymptotic expansions for the ."/ mixed power-exponential moments 0 ij Œ; n are given in Lemmas 5.3.5 and 5.3.6.

341

5.5 Asymptotics of quasi-stationary distributions ."/

Similar asymptotic expansions can be given for the expectations !ij Œ; n. The following representation can be written for l; j ¤ 0: Z

j."/ ^."/ 0

0

e s .."/ .s/ D l/ ds Z

D

."/

1 0

X

e s .."/ .s/ D l/ ds C

."/

.."/ . 1 / D i 0 /

i 0 ¤0;j ."/

 e 1



Z

."/ j."/ ^."/ 0 1

0

e s .."/ . 1."/ C s/ D l/ ds:

(5.5.50)

We will use the following simple relation for "  "06 ,   ˇ0 and i; l ¤ 0: Z Ei

."/

1 0

."/

e s .."/ .s/ D l/ ds D ı.i; l/i Œ; 0 ( D .0/

."/

ı.i; l/.Ei e 1  1/=

for  ¤ 0,

."/ ı.i; l/Ei 1

for  D 0.

(5.5.51)

."/

Note again that X1 X1 for "  "07 . By taking the expectation in the left and the right-hand sides of (5.5.50) and using the Markov property of the semi-Markov process ."/ .t / at the time of the first jump, as well as relation (5.5.51), this representation permits to readily show that the moment ."/ generating functions !ij l Œ; 0, i ¤ 0, satisfy the following system of linear equations .0/

for every "  "06 ,   ˇ 0 , l ¤ 0, and j 2 X1 : ."/

."/

!ij l Œ; 0 D 'i Œ; 0ı.i; l/ C

X

."/

."/

i ¤ 0:

pi i 0 ./ !i 0 j l Œ; 0;

(5.5.52)

i 0 ¤0;j

In virtue of the remarks above, we can differentiate equations (5.5.52) with respect to the variable  and get the following system of linear equations for every "  "07 , .0/   ˇ0 , and n D 0; 1; : : :, j 2 X1 : ."/

."/

!ij l Œ; n D #ij l Œ; n C

X

."/

."/

pi i 0 ./!i 0 j l Œ; n;

i ¤ 0;

(5.5.53)

i 0 ¤0;j

where ."/

."/

#ij l Œ; n D ı.i; l/'i Œ; n C

n X X i 0 ¤0;j

mD1

."/

."/

Cnm pi i 0 Œ; m!i 0 j l Œ; n  m:

(5.5.54)

342

5

Nonlinearly perturbed semi-Markov processes

These systems can be rewritten in a matrix form. For   ˇ 0 , "  "07 , l ¤ 0 and .0/ n D 0; 1; : : :, l ¤ 0, and j 2 X1 , consider the vectors 2

3 ."/ Œ; n !1j l 6 7 ."/ :: 7; ¨jl Œ; n D 6 : 4 5 ."/ !Nj l Œ; n

2

3 ."/ Œ; n #1j l 6 7 ."/ :: 7: ª j l Œ; n D 6 : 4 5 ."/ #Nj l Œ; n

Then, the linear systems (5.5.52) can be rewritten in the following form for every .0/   ˇ 0 , "  "07 , l ¤ 0 and j 2 X1 : ."/ ."/ ."/ ¨."/ jl Œ; 0 D ®l Œ; 0 C j P Œ; 0 ¨jl Œ; 0;

(5.5.55)

and for every "  "07 ,   ˇ 0 , n D 0; 1; : : :, l ¤ 0, and j 2 X1.0/ , (5.5.53) becomes ."/

."/

."/

¨jl Œ; n D ª jl Œ; n C j P."/ ./ ¨jl Œ; n: ."/

."/

(5.5.56) .0/

It is convenient to also define #ij l Œ; 0 D ı.i; l/'l ./, i; l ¤ 0, j 2 X1 ."/ ª jl Œ; 0

."/ ®l Œ; 0,

and,

.0/ X1 .

D l ¤ 0, j 2 respectively, In this case, systems (5.5.53) and (5.5.56) coincide with systems (5.5.52) and (5.5.55), respectively, for n D 0. Recall also that !ij."/l ./ D !ij."/l Œ; 0, l ¤ 0, j 2 X1.0/ . Define the vectors ."/ .0/ ¨."/ j l ./ D ¨j l Œ; 0, l ¤ 0, j 2 X1 . The systems of linear equations (5.5.56) are systems of -hitting type with the same coefficient matrix j P."/ ./. Under conditions A2 , T1 , E2 , C12 , C13 , these systems have unique solutions for every "  "07 ,   ˇ 0 , n D 0; 1; : : :, l ¤ 0, and j 2 X1.0/ . These solutions have the following forms: ."/ 1 ."/ ¨."/ jl Œ; n D ŒI  j P ./ ª jl Œ; n: .0/

(5.5.57)

For given l ¤ 0, j 2 X1 , systems (5.5.56) have the same coefficient matrix ."/ ."/ j P ./ but different inhomogeneous terms ª jl Œ; n for n D 0; 1; : : : . These systems should be solved recursively. First, system (5.5.56) should be solved for n D 0. Second, we solve system ."/ (5.5.56) for n D 1. Note that the expressions for inhomogeneous terms ij l Œ; 1 D P ."/ ."/ ."/ ı.i; l/'i Œ; 1 C l¤0;j pi i 0 Œ; 1!i 0 j l Œ; 0, i ¤ 0, given in (5.5.54), include the solutions !ij."/l Œ; 0, i ¤ 0, of systems (5.5.56) for n D 0.

5.5 Asymptotics of quasi-stationary distributions

343

This recursive procedure is to be repeated for n D 1; : : : . The expressions for the inhomogeneous terms #ij."/l Œ; n, given in (5.5.54), include the solutions !ij."/l Œ; m, i ¤ 0, of systems (5.5.56) for m D 0; 1; : : : ; n  1. Formula (5.5.54) can be rewritten in the following vector form for every   ˇ 0 , .0/ "  "07 , and n D 0; 1; : : :, l ¤ 0, j 2 X1 : ."/ ª jl Œ; n

D

."/ ®l Œ; n

C

n X

."/

Cnm j P."/ Œ; m¨jl Œ; n  m:

(5.5.58)

mD1

Using formula (5.5.58) we can rewrite formula (5.5.56) in a form that indicates the recurrence procedure used for calculating solutions of the linear system (5.5.56) for .0/   ˇ 0 , "  "07 , and n D 0; 1; : : :, l ¤ 0, j 2 X1 , ."/

."/

¨jl Œ; n D ŒI  j P."/ ./1 ª jl Œ; n ."/

D ŒI  j P."/ ./1 ®l Œ; n C

n X

."/

Cnm ŒI  j P."/ ./1 j P."/ Œ; m¨jl Œ; n  m:

(5.5.59)

mD1

We are now in a position to derive asymptotic expansions for the functions n D 0; 1; : : : ; k, l ¤ 0, j 2 X1.0/ . The following lemmas are corollaries of Lemma 5.3.4 applied to the vector-valued .0/ functions ¨."/ j Œ; n for l ¤ 0, j 2 X1 and n D 0; 1; : : : ; k. Their proofs are similar to those given for Lemmas 5.3.5 and 5.3.6.

."/ Œ; n, !ijl

.k/ hold, and, for some   ˇ 0 , Lemma 5.5.7. Let conditions A2 , T2 , E2 , C12, C13 , P18 .;k/ .0;kC1/ either condition P29 , if   ˇ 0 ;  ¤ 0, or P29 , if  D 0, be satisfied. Then

(i) For every l ¤ 0; j 2 X1.0/ , the vector ¨."/ jl Œ; 0 admits an asymptotic expansion, ."/

.0/

¨jl Œ; 0 D ¨jl Œ; 0 C cjl Œ; 1; 0" C    C cjl Œ; k; 0"k C o."k /; (5.5.60) where cjl Œ; r; 0, r D 1; : : : ; k, are finite vectors. (ii) The vector coefficients cjlŒ; r; 0 are given by the recurrence formulas .0/ .0/ cjl Œ; 0; 0 D ¨jl Œ; 0 D ŒI  j P.0/ ./1 ®l Œ; 0 and, for r D 0; : : : ; k, cjl Œ; r; 0 D j UŒ; 0.wl Œ; r; 0 C

r X qD1

j EŒ; q; 0cjl Œ; r

 q; 0/:

(5.5.61)

344

5

Nonlinearly perturbed semi-Markov processes

(iii) The vector coefficients cjlŒ; r; 0 can also be calculated by the following formulas for r D 0; : : : ; k: r X

cjlŒ; r; 0 D

j UŒ; p wl Œ; r

 p; 0:

(5.5.62)

pD0 .0/

."/

(iv) For every l ¤ 0, j 2 X1 , and n D 0; : : : ; k, the vector ¨jl Œ; n admits the following asymptotic expansion: ."/

.0/

¨jl Œ; n D ¨jl Œ; n C cj l Œ; 1; n" C    C cjl Œ; k  n; n"kn C o."kn /;

(5.5.63)

where cjl Œ; r; n, r D 1; : : : ; k  n, n D 0; : : : ; k, are finite vectors. (v) The vector coefficients cjl Œ; r; n are given by the recurrence formulas .0/ .0/ .0/ cjl Œ; 0; 0 D ¨jl Œ; 0 D j UŒ; 0ª jl Œ; 0 D ŒI  j P.0/ ./1 ¥l Œ; 0 and, for r D 0; : : :, k  n, and and sequentially for n D 0; : : : ; k, cjlŒ; r; n D j UŒ; 0.wl Œ; r; n C

n X

Cnm

mD1

C

r X

r X

j EŒ; q; mcjl Œ; r

 q; n  m

qD0

Ej Œ; p; 0cjl Œ; r  p; n/:

(5.5.64)

pD1

(vi) The vector coefficients cjl Œ; r; n can also be calculated by the recurrence formulas for r D 0; : : : ; k  n and sequentially for n D 0; : : : ; k, cjlŒ; r; n D

r X

j UŒ; qwl Œ; r

qD0

C

n X mD1

Cnm

r X qD0

 q; n

0 @

q X

1 j UŒ; pj EŒ; q

 p; mA

pD0

 cjl Œ; r  q; n  m:

(5.5.65)

Remark 5.5.2. Note that the asymptotic expansions of the order k for vectors .;k/ .0;kC1/ , if  D 0. ¨."/ jl Œ; n require condition P29 , if  ¤ 0, and condition P29 Remark 5.5.3. Note that condition I4 is not required in Lemma 5.5.7.

345

5.5 Asymptotics of quasi-stationary distributions

Formula (5.5.28) implies that, for n D 0; 1; : : :, i; j; l ¤ 0, ."/

!ij Œ; n D

X

."/

(5.5.66)

!ij l Œ; n:

l¤0 .0/

For   ˇ0 , "  "07 , l ¤ 0, and n D 0; 1; : : : ; k, l ¤ 0, and j 2 X1 , introduce the vectors X cj Œ; r; n D cjlŒ; r; n: (5.5.67) l¤0

For r D 0; : : : ; k  n, n D 0; : : : ; k, also define the vectors 2 ."/ 3 !1j Œ; n 6 7 :: 6 7; ¨."/ : j Œ; n D 4 5 ."/ !Nj Œ; n and 3 c1j Œ; r; n 7 6 :: cj Œ; r; n D 4 5; : cNj Œ; r; n 2

3 c1j l Œ; r; n 7 6 :: cjlŒ; r; n D 4 5: : cNj l Œ; r; n 2

."/

Relation (5.5.66) permits to obtain asymptotic expansions for the vectors ¨j Œ; n

by using asymptotic expansions for the vectors ¨."/ jl Œ; n given in Lemma 5.5.7. These .0/

expansions take the following form for "  "06 ,   ˇ0 , j 2 X1 , and n D 0; 1; : : : ; k: .0/ kn C o."kn /: ¨."/ j Œ; n D ¨j Œ; n C cj Œ; 1; n" C    C cj Œ; k; n"

(5.5.68)

.0/

Usually, it is also convenient to define the vectors cj Œ; 0; n D ¨j Œ; n, n D 0; : : : ; k.

5.5.8 Asymptotic expansions for quasi-stationary distributions of perturbed semi-Markov processes We are now in a position to give asymptotic expansions for quasi-stationary distributions and related quasi-stationary limits of perturbed semi-Markov processes. The process ."/ .t /, t  0, is a regenerative process with absorption, and the times ."/ of return to the initial state are the corresponding regeneration times. Also, 0 is a regenerative stopping time which regenerate together with the process ."/ .t /, t  0, at the times of return. Therefore, we can use theorems given in Section 3.5.

346

5

Nonlinearly perturbed semi-Markov processes

The corresponding imbedding procedure is described in Section 4.2. In all “imbedding” calculations a one-state set A D flg and a recurrent-without-absorption state j 2 X1.0/ are used for constructing the corresponding regeneration cycles. The moment generating functions  ."/ ./, Q ."/ ./, and ! ."/ .; r; A/ are regarded ."/ ."/ ."/ as the moment generating functions 0 jj ./, 0 ij ./, and !jj l ./, respectively. The respective asymptotic expansions for the moment generating functions ."/ ."/ ."/ ."/  0 jj ./, 0 ij ./, and !jj l ./ are given in terms of the transition probabilities pir ."/

of imbedded Markov chains, and the moment generation functions pir Œ; n of transition times. These conditions are used in Lemmas 5.3.5, 5.3.6, and 5.5.7, where the ."/ corresponding perturbation expansions for the moment generation functions 0 jj ./, ."/ 0 ij ./,

."/ and !jj ./ are obtained. l All other conditions used in Theorems 3.5.2–3.5.4 are checked in the proofs of Theorems 5.4.1–5.4.4. Since any recurrent-without-absorption state j 2 X1.0/ can be used in the construction of corresponding regeneration cycles, the corresponding quasi-stationary distribu.0/ tions do not depend on the choice of the state j 2 X1 . Taking into account the above remarks, we formulate the following four theorems without explicit proofs. The next theorem is a variant of Theorem 5.4.1. .0/

.k/ ;k/ hold, and either condition P. , Theorem 5.5.2. Let conditions A2 , I4 T2 , E2 , P18 29 .0;kC1/ .0/ .0/ if  > 0 or P29 , if  D 0, be satisfied. Then, the quasi-stationary limits ."/ ."/ ."/ ."/ ."/ ."/ .0/ Qj l . / D !jj l . /= 0 jj . /, l ¤ 0, j 2 X1 , have the following asymptotic .0/

expansion for every l ¤ 0 and j 2 X1 : .0/

Qj."/ .."/ / l

D

!jj l ..0/ / C fjj0 l Œ.0/ ; 1" C    C fjj0 l Œ.0/ ; k"k C o."k / .0/ .0/ 0 jj . /

C fjj00 Œ.0/ ; 1" C    C fjj00 Œ.0/ ; k"k C o."k /

..0/ / C fj l Œ.0/ ; 1" C    C fj l Œ.0/ ; k"k C o."k /; D Qj.0/ l where the coefficients fjj0 l Œ.0/ ; n and fjj00 Œ.0/ ; n are given by the formulas .0/

fjj0 l Œ.0/ ; 0 D !jj l ..0/ / D cjj l Œ.0/ ; 0; 0; fjj0 l Œ.0/ ; 1 D cjj l Œ.0/ ; 1; 0 C cjj l Œ.0/ ; 0; 1a1 ; .0/

fjj00 Œ.0/ ; 0 D 0 jj ..0/ / D 0 bjj Œ.0/ ; 0; 1; fjj00 Œ.0/ ; 1 D 0 bjj Œ.0/ ; 1; 1 C 0 bjj Œ.0/ ; 0; 2a1 ;

(5.5.69)

347

5.5 Asymptotics of quasi-stationary distributions

and, for n D 0; : : : ; k, fjj0 l Œ.0/ ; n D cjj l Œ.0/ ; n; 0 C

n X

cjj l Œ.0/ ; n  q; 1aq

qD1

C

X

n X

X

cjj l Œ.0/ ; n  q; m 

2mn qDm

q1 Y

n

app =np Š

(5.5.70)

n1 ;:::;nq1 2Dm;q pD1

and fjj0 Œ.0/ ; n D 0 bjj Œ.0/ ; n; 1 C

n X

0 bjj Œ

.0/

; n  q; 2aq

qD1

C

X

n X

0 bjj Œ

.0/

X

; n  q; m C 1 

2mn qDm

q1 Y

n

app =np Š; (5.5.71)

n1 ;:::;nq1 2Dm;q pD1

and the coefficients fj l Œ.0/ ; n are given by the recurrence formulas fj l Œ.0/ ; 0 D Qj.0/ ..0/ / D fjj0 l Œ.0/ ; 0=fjj00 Œ.0/ ; n and, for n D 0; : : : ; k, l   n1 X fjj00 Œ.0/ ; n  q fj l Œ.0/ ; q =fjj00 Œ.0/ ; 0: (5.5.72) fj l Œ.0/ ; n D fjj0 l Œ.0/ ; n  qD0 .0/

For j 2 X1 , consider the quasi-stationary quantities X ."/ Qj."/ .."/ / D Qjr .."/ /: r ¤0

Theorem 5.5.2 gives the following asymptotic expansion for these quantities; it is a direct corollary of the asymptotic expansion (5.5.69), ."/

.0/

Qj .."/ / D Qj ..0/ / C fj Œ.0/ ; 1" C    C fj Œ.0/ ; k"k C o."k /;

(5.5.73)

where the coefficients fj Œ.0/ ; n are given by the recurrence formulas fj Œ.0/ ; 0 D .0/ Qj ..0/ / and, for n D 0; : : : ; k, fj Œ.0/ ; n D

X

fjr Œ.0/ ; n:

r ¤0

The following theorem is a modification of Theorem 5.4.2.

(5.5.74)

348

5

Nonlinearly perturbed semi-Markov processes ..0/ ;k/

.k/

Theorem 5.5.3. Let conditions A2 , I4 T2 , E2 , P18 hold, and either condition P29 , .0;kC1/ .0/ .0/ if  > 0 or P29 , if  D 0, be satisfied. Then, the quasi-stationary limits ."/ ."/ ."/ ."/ QQ il .."/ / D 0 ij .."/ /!jj l .."/ /= 0 jj .."/ /, i; l ¤ 0, which do not depend on .0/

the choice of the state j 2 X1 , have the following asymptotic expansion for every .0/ i; l ¤ 0 and j 2 X1 : ."/ .0/ QQ il .."/ / D QQ il ..0/ / C hil Œ.0/ ; 1" C    C hil Œ.0/ ; k"k C o."k /;

(5.5.75)

where the coefficients hil Œ.0/ ; n are given by the formulas hil Œ.0/ ; 0 D QQ il.0/ ..0/ / D fij000 Œ.0/ ; 0fj l Œ.0/ ; 0; hil Œ.0/ ; 1 D fij000 Œ.0/ ; 0fj l Œ.0/ ; 1 C fij000 Œ.0/ ; 1fj l Œ.0/ ; 0; and, for n D 0; : : : ; k, hil Œ.0/ ; n D

n X

fij000 Œ.0/ ; qfj l Œ.0/ ; n  q

(5.5.76)

qD0

and fij000 Œ.0/ ; n n X

D 0 bij Œ.0/ ; n; 0 C

0 bij Œ

.0/

; n  q; 0aq

qD1

C

X

n X

X

.0/ 0 bij Π; n  q; m 

2mn qDm

q1 Y

n

app =np Š:

(5.5.77)

n1 ;:::;nq1 2Dm;q pD1

The following theorem is a version of Theorem 5.4.3. .k/

..0/ ;k/

, Theorem 5.5.4. Let conditions A2 , I4 T2 , E2 , P18 hold, and either condition P29 .0;kC1/ .0/ .0/ if  > 0 or P29 , if  D 0, be satisfied. Then, the quasi-stationary probabili."/ ."/ ."/ ."/ ."/ .0/ ties l . / D Qj l . /=Qj .."/ /, l ¤ 0, j 2 X1 , which do not depend on the .0/

choice of the state j 2 X1 , have the following asymptotic expansion for every l ¤ 0 .0/ and j 2 X1 : .0/

l."/ .."/ /

D

Qj l ..0/ / C fj l Œ.0/ ; 1" C    C fj l Œ.0/ ; k"k C o."k / .0/

Qj ..0/ / C fj Œ.0/ ; 1" C    C fj Œ.0/ ; k"k C o."k /

D  .0/ ..0/ / C gl Œ.0/ ; 1" C    C gl Œ.0/ ; k"k C o."k /;

(5.5.78)

349

5.5 Asymptotics of quasi-stationary distributions

where the coefficients gl Œ.0/ ; n are given by the recurrence formulas gl Œ.0/ ; 0 D  .0/ ..0/ / D fj l Œ.0/ ; 0=fj Œ.0/ ; 1; .0/

fj l Œ.0/ ; 0 D Qj l ..0/ /; fj Œ.0/ ; 0 D Qj.0/ ..0/ /; and, for n D 0; : : : ; k, gl Œ.0/ ; n D .fj l Œ.0/ ; n 

n1 X

fj Œ.0/ ; n  qgl Œ.0/ ; q/=fj Œ.0/ ; 0:

(5.5.79)

qD0

The next theorem is a variant of Theorem 5.4.4. .0/

.k/ ;k/ Theorem 5.5.5. Let conditions A2 , I4 T2 , E2 , P18 hold, and either condition P. , 29 .0;kC1/ .0/ .0/ if  > 0 or P29 , if  D 0, be satisfied. Then, the quasi-stationary probabili."/ ."/ ."/ ."/ ties l."/ .."/ / D !jj . /=!jj . /, l ¤ 0, j 2 X1.0/ , which do not depend on the l

choice of the state j 2 X1.0/ , have the following asymptotic expansion for every l ¤ 0 and j 2 X1.0/ : .0/

."/ l .."/ /

D

!jj l ..0/ / C fjj0 l Œ.0/ ; 1" C    C fjj0 l Œ.0/ ; k"k C o."k / .0/

!jj ..0/ / C fjj0 Œ.0/ ; 1" C    C fjj0 Œ.0/ ; k"k C o."k /

D  .0/ ..0/ / C gl Œ.0/ ; 1" C    C gl Œ.0/ ; k"k C o."k /;

(5.5.80)

where the coefficients fjj0 l Œ.0/ ; n and fjj0 Œ.0/ ; n are given by the formulas .0/

fjj0 l Œ.0/ ; 0 D !jj l ..0/ / D cjj l Œ.0/ ; 0; 0; fjj0 l Œ.0/ ; 1 D cjj l Œ.0/ ; 1; 0 C cjj l Œ.0/ ; 0; 1a1 ; .0/ .0/ . / D cjj Œ.0/ ; 0; 0; fjj0 Œ.0/ ; 0 D !jj

fjj0 Œ.0/ ; 1 D cjj Œ.0/ ; 1; 0 C cjj Œ.0/ ; 0; 1a1 ; and, for n D 0; : : : ; k, fjj0 l Œ.0/ ; n D cjj l Œ.0/ ; n; 0 C

n X

cjj l Œ.0/ ; n  q; 1aq

qD1

C

X

n X

2mn qDm

cjj l Œ.0/ ; n  q; m 

X

q1 Y

n1 ;:::;nq1 2Dm;q pD1

n

app =np Š;

(5.5.81)

350

5

Nonlinearly perturbed semi-Markov processes

and fjj0 Œ.0/ ; n D cjj Œ.0/ ; n; 0 C

n X

cjj Œ.0/ ; n  q; 1aq

qD1

C

X

n X

cjj Œ.0/ ; n  q; m 

2mn qDm

X

q1 Y

n

app =np Š;

(5.5.82)

n1 ;:::;nq1 2Dm;q pD1

and coefficients gl Œ.0/ ; n are given by the recurrence formulas gl Œ.0/ ; 0 D  .0/ ..0/ / D fjj0 l Œ.0/ ; 0=fjj0 Œ.0/ ; 0, and, for n D 0; : : : ; k, gl Œ.0/ ; n D .fjj0 l Œ.0/ ; n 

n1 X

fjj0 Œ.0/ ; n  qgl Œ.0/ ; q/=fjj0 Œ.0/ ; 0: (5.5.83)

qD0

It is useful to note that despite the difference in the algorithms, coefficients gl Œ.0/ ; n, n D 1; : : : ; k, in both asymptotic expansions (5.5.78) and (5.5.80) coincide. In conclusion we would like to remark that the theorems formulated above cover both the pseudo- and quasi-stationary models for .0/ D 0 and .0/ > 0, respectively. However, even in the pseudo-stationary case, Theorems 5.5.2, 5.5.3, 5.5.4, and 5.5.5 differ from Theorem 5.5.1. Indeed, the former four theorems allow for an absorption of pre-limit perturbed processes, while this is impossible for pre-limit perturbed processes in the Theorem 5.5.1.

5.6 Nonlinearly perturbed continuous-time Markov chains In this section we consider a particular but important case where nonlinearly perturbed continuous-time Markov chains with absorption.

5.6.1 Perturbed continuous-time Markov chains Let ."/ .t /, t  0 be a continuous-time homogeneous Markov chain for every "  0, with the phase space X D f0; : : : ; N g and a generator Q."/ that has the following form: 3 2 ."/ ."/ ."/ q01 : : : q0N q0 6 ."/ ."/ 7 7 6 q10 q1."/ : : : q1N ."/ 7; 6 Q D6 : : : : :: :: :: 7 5 4 :: ."/ ."/ ."/ qN 0 qN1 : : : qN P ."/ ."/ ."/ where (a) 0  qij < 1, i ¤ j , i; j 2 X, (b) qi D j ¤i qij , i 2 X.

5.6

Nonlinearly perturbed continuous-time Markov chains

351

We consider the case where 0 is an absorption state, so that the following condition holds: ."/

A6 : q0 D 0 for any "  0. We also assume that the following condition is satisfied: ."/ .0/ ! qij as " ! 0 for i ¤ 0; j 2 X. T3 : qij

Condition T3 allows to consider the Markov chain ."/ .t /, t  0, for " > 0 as a perturbed version of the Markov chain .0/ .t /, t  0. If condition T3 holds, then, for i ¤ 0, ."/

qi

.0/

! qi

as " ! 0:

(5.6.1)

In order to exclude the existence of absorption states j ¤ 0, we also assume that the following condition is satisfied: .0/

I6 : qj

> 0; j ¤ 0. ."/

Conditions I6 and T3 imply that (a) there exists "0 > 0 such that qj > 0, j ¤ 0, for "  "0 . The Markov chain ."/ .t /, t  0 is a particular case of the semi-Markov process with absorption, for "  "0 . It has the transition probabilities ( ."/ Qij .t /

D

."/

."/

pij .1  e qi

t

/ if i ¤ 0, j ¤ i , j 2 X; if i ¤ 0, j D i;

0

(5.6.2)

where ( ."/ pij

D

."/

."/

qij =qi

if i ¤ 0, j ¤ i , j 2 X,

0

if i ¤ 0, j D i .

(5.6.3)

In this case, the distributions of transition times are exponential, with the parameters that do not depend on j 2 X. They have the following form for i ¤ 0, j 2 X: ."/

."/

."/

Fij .t / D Fi .t / D .1  e qi

t

/:

(5.6.4)

Conditions A6 , T3 , and I6 imply and replace, respectively, conditions A2 , T1 , and I4 .

Conditions E2 , E02 and E002 do not change their formulations.

352

5

Nonlinearly perturbed semi-Markov processes

5.6.2 Regularisation of continuous-time Markov chains The standard definition of continuous-time Markov chain excludes possibility for a Markov chain to have virtual transitions i ! i . In some cases, it is convenient to allow such transitions. This can be achieved by adding some additional intensities qi  0, i ¤ 0 for such transitions and by replacing the Markov chain ."/ .t /, t  0, by a semi-Markov process Q ."/ .t /, t  0, with the transition probabilities ( ."/ ."/ .1  e qQi t / if i ¤ 0, j ¤ i , j 2 X; pQij ."/ QQ ij .t / D (5.6.5) 0 if i ¤ 0, j D i; where qQi."/ D qi C qi."/ ; and

( ."/ pQij

D

."/

."/

qij =qQ i

qi =qQ i."/

i ¤ 0:

(5.6.6)

if i ¤ 0, j ¤ i , j 2 X, if i ¤ 0, j D i .

(5.6.7)

Simple calculation shows that the processes ."/ .t /, t  0, by a semi-Markov process Q ."/ .t /, t  0, have the same finite-dimensional distributions, i.e., for any i; i1 ; : : : ; in ¤ 0 and 0  t1      tn < 1, n D 1; 2; : : :, Pi f."/ .tr / D ir ; r D 1; : : : ; ng D Pi fQ ."/ .tr / D ir ; r D 1; : : : ; ng:

(5.6.8)

Relation (5.6.8) obviously implies that the semi-Markov process Q ."/ .t /, t  0 is, in fact, a homogeneous Markov process. This process can replace the Markov chain ."/ .t /, t  0 in any quasi-stationary asymptotic relations.

5.6.3 Moment conditions The distributions of transition times have finite power moments of all orders, do not depend on j 2 X, and are given by the following explicit formula for n D 0; 1; : : :, i ¤ 0, j 2 X and "  "0 : Z 1 ."/ ."/ ."/ ."/ ."/ s n qi e qi s ; ds D nŠ.qi /n < 1: (5.6.9) mij Œn D mi Œn D 0

."/

The moment generating functions of random variables, 1 , also do not depend on j 2 X and are given by the following explicit formula for n D 0; 1; : : :, i ¤ 0, j 2 X: Z 1 ."/ ."/ ."/ ."/ s n e s qi e qi s ds ij Œ; n D i Œ; n D D

8 < :

0

."/ nŠqi ."/ .qi  /nC1

1

."/

< 1 if  < qi ; ."/

if   qi .

(5.6.10)

5.6

353

Nonlinearly perturbed continuous-time Markov chains

Condition C12 holds for any 0 < ı <  , where .0/

.0/

 D min.q1 ; : : : ; qN /:

(5.6.11)

Condition C13 remains the same. The pseudo-stationary case, where the characteristic root .0/ D 0, corresponds P .0/ with condition A3 that takes the form i2X .0/ qi0 D 0. 1 In this case, as was shown in Lemma 4.5.8, condition C13 automatically holds. The quasi-stationary case, where the characteristic root .0/ > 0, corresponds with P .0/ the assumption that condition A3 does not hold, i.e., i2X .0/ qi0 > 0. 1 In this case, condition G1 holds with the parameter .0/

.0/

 D min.qi W i 2 X1 /:

(5.6.12)

Condition G2 (a) holds if  D  . Indeed, in this case, for  <  and i ¤ 0, lim

"!0

."/ i Œ; 0

D lim

"!0

qi."/ qi."/  

D

.0/ i Œ; 0

D

qi.0/ qi.0/  

< 1:

(5.6.13)

Condition G2 (b) remains. As was shown in Lemma 4.5.10, condition C13 is implied by conditions G1 and G2 . In this case, the characteristic root 0 < .0/ <  . Fortunately, as was mentioned in Subsection 4.5.5, in the quasi-stationary case, condition E02 usually holds, i.e. the set of non-recurrent-without-absorption states .0/ X2 D ¿. In this case, conditions  D  and G2 , which is an analogue of C13 (b), automatically holds.

5.6.4 Perturbation conditions The perturbation condition that implies and replaces the corresponding perturbation conditions on the transition probabilities of imbedded Markov chain, the power moments, and the mixed power-exponential moments of distributions of transition times, and was used in Chapter 5, takes the following form: .k/

."/

.0/

P30 : qij D qij C qij Œ1" C    C qij Œk"k C o."k / for i ¤ 0, j 2 X, where jqij Œrj < 1, r D 0; : : : ; k, i ¤ 0, j 2 X. .0/

It is convenient to also set qij Œ0 D qij , i ¤ 0, j 2 X. The parameter k in .k/

condition P30 can take any non-negative integer value. .k/ Note also that condition P30 implies that condition T3 holds. .k/ The following two lemmas show that condition P30 implies the perturbation con.k/ dition P18 for transition probabilities of an imbedded Markov chain.

354

5

Nonlinearly perturbed semi-Markov processes .k/

Lemma 5.6.1. Let conditions I6 , and P30 hold. Then the following expansion holds for every i ¤ 0, j 2 X: ."/

qi

D qi Œ0 C qi Œ1" C    C qi Œk"k C o."k /;

(5.6.14)

where qi Œ0 D qi.0/ and, for r D 0; : : : ; k, qi Œr D

X

qij Œr;

r D 0; : : : ; k:

(5.6.15)

j ¤i

Proof. The lemma is a direct corollary of formula (5.6.3), relation (b), and Lemma ."/ 8.1.1. According to relation (b), the intensities qi , i ¤ 0, are obtained as sums of the ."/ intensities qij . Thus, the asymptotic expansion (5.6.14) is a corollary of statement (ii) of Lemma 8.1.1. .k/ Lemma 5.6.2. Let conditions A5 , I6 , and P30 hold. Then we have the following expansion for every i ¤ 0, j 2 X: ."/

.0/

pij D pij C eij Œ1; 0" C    C eij Œk; 0"k C o."k /;

(5.6.16)

where the coefficients eij Œr; 0, r D 0; : : : ; k, are given by the formulas eij Œ0; 0 D .0/ pij D qij Œ0=qi Œ0, eij Œ1; 0 D .qij Œ1  eij Œ0; 0qi Œ1/=qi Œ0 and, in general, for r D 0; : : : ; k, eij Œr; 0 D .qij Œr 

rX 1

eij Œm; 0qi Œr  m/=qi Œ0:

(5.6.17)

mD0

Proof. The lemma immediately follows from formula (5.6.3), relation (b), and Lemma ."/ 8.1.1. The transition probabilities pij are given by formula (5.6.3) as a quotient of

."/ and qi."/ . Therefore, asymptotic expansion (5.6.16) can be the intensity functions qij used for these transition probabilities, due to statement (v) of Lemma 8.1.1.

The explicit formula (5.6.9) yields expansions for the power moments of transi.k/ tion times and permits to prove that condition P30 implies to hold the perturbation Q

.k/ .k/ conditions P20 , P21 , and P.22k/ for the power moments of transition times. .k/

Lemma 5.6.3. Let conditions A5 , I6 , and P30 be satisfied. Then the following expansion takes place for every i ¤ 0, j 2 X and n  1: .0/ k k m."/ i Œn D mi Œn C vi Œ1; n" C    C vi Œk; n" C o." /;

(5.6.18)

5.6

355

Nonlinearly perturbed continuous-time Markov chains

where the coefficients vi Œr; n, r D 0; : : : ; k, n  1, are given by the formulas n vi Œ0; n D m.0/ ij Œn D nŠ=.qi Œ0/ , n  1, and, for r D 1; : : : ; k and n  1, X r nŠ vi Œr  l; n  vi Œr; n D  .qi Œ0/n lD1

X

l Y

 qi Œp =np Š ; np

(5.6.19)

 pD0 n0 ;:::;nl 2Dn;l

 is the set of all nonnegative, integer solutions of the system where Dn;l

n0 C n1 C    C nl D n;

n1 C    C lnl D l:

(5.6.20)

Proof. It follows from the explicit formula (5.6.9) and Lemma 8.1.1 that the power moments m."/ i Œn, n  1, can be expanded in .0; k/-asymptotic expansions. To see this, one must first apply statement (iv) of this lemma to get an .0; k/-asymptotic ."/ expansion for the function .qi /1 and then apply n  1 times the product rule, given in statement (iii), to this function, in order to get the .0; k/-asymptotic expansion for ."/ function .qi /n . However, it is better to find the coefficients in expansion (5.6.18) by equating the coefficients of "r , r D 0; 1; : : : ; k, in the left and right hand-sides of the formal asymptotic identity .vi Œ0; n C vi Œ1; n" C    /.qi Œ0 C qi Œ1" C    /n D nŠ ;

(5.6.21)

which immediately yields formulas (5.6.19). .k/

.kQ /

Consider the functions used in the perturbation conditions P20 and P22 , ."/ ."/ ."/ Œn D pij mi Œn; pij

n  1;

i ¤ 0;

j 2 X:

.k/

Lemma 5.6.4. Let conditions A5 , I6 , and P30 hold. Then there is an expansion that holds for every i ¤ 0, j 2 X and n  1, ."/ .0/ pij Œn D pij Œn C eij Œ1; n" C    C eij Œk; n"k C o."k /;

(5.6.22)

where the coefficients eij Œr; n, r D 0; : : : ; k, n  1, are given by the formulas .0/ .0/ .0/ eij Œ0; n D pij Œn D pij mi Œn D eij Œ0; 0vi Œ0; n, n  1, and, for r D 0; : : : ; k and n  1, 8 for r D 0; : : : ; k; n D 0, < eij Œr; 0 r P (5.6.23) eij Œr; n D : eij Œm; 0vi Œr  m; n for r D 0; : : : ; k; n  1: mD0

Proof. The proof follows from statement (iii) of Lemma 8.1.1 applied to the product ."/ ."/ pij mi Œn.

356

5

Nonlinearly perturbed semi-Markov processes .k/

Remark 5.6.1. According to Lemma 5.6.4, condition P30 implies that the perturba.kQ / tion condition P holds for any nQ  1 and 0  kQn  k, n D 0; : : : ; n. Q 22

The explicit formula (5.6.10) yields expansions for the mixed power-exponential .k/ moments of transition times and permits to prove that if condition P30 holds, then .;k/

.;k/

.;kQ /

.;k/

.;k/

implies the perturbation conditions P23 – P26 , P27 , P28 , and P29 for the mixed power-exponential moments of transition times.

also hold

.k/

Lemma 5.6.5. Let conditions A5 , I6 , and P30 hold. Then we have the following ex.0/ pansion for every i ¤ 0, j 2 X, n  1, and  < qi D qi Œ0: ."/ i Œ; n

D

.0/ i Œ; n C

vi Œ; 1; n" C    C vi Œ; k; n"k C o."k /;

(5.6.24)

where the coefficients vi Œ; r; n, r D 0; : : : ; k, n  1, are given by the formulas .0/ vi Œ; 0; n D ij Œ; n D nŠ=.qi Œ0  /nC1 , n  1, and, for r D 1; : : : ; k and n  1, vi Œ; r; n D

X r nŠ .n C 1/Š  vi Œ; r  l; n .qi Œ0  /nC1 .qi Œ0  /nC1 lD1



X  n0 ;:::;nl 2DnC1;l

 l .qi Œ0  /n0 Y qi Œpnp ; n0 Š np Š

(5.6.25)

pD1

 where DnC1;l is the set of all nonnegative, integer solutions of the system

n0 C n1 C    C nl D n C 1;

n1 C    C lnl D l:

(5.6.26)

Proof. It follows from the explicit formula (5.6.10) and Lemma 8.1.1 that the power ."/ moments i Œ; n, n  1, can be expanded in .0; k/-asymptotic expansions. To see this, one must first apply statement (iv) of this lemma to get an .0; k/-asymptotic ."/ expansion for the function .qi  /1 , and then apply n C 1 times the product rule in statement (iii) to this function for obtaining an .0; k/-asymptotic expansion for the function qi."/ .qi."/  /n1 . However, it is easier to find the coefficients in expansion (5.6.24) by equating the coefficients of "r , r D 0; 1; : : : ; k, in the left and right hand-sides of the formal asymptotic identity



nC1 vi Œ; 0; n C vi Œ; 1; n" C    .qi Œ0  / C qi Œ1" C   

D nŠ qi Œ0 C qi Œ1" C    ;

which at once yields formulas (5.6.25).

(5.6.27)

5.6

357

Nonlinearly perturbed continuous-time Markov chains .;kQ /

.k/

Remark 5.6.2. Condition P30 implies that the perturbation condition P27 any 0  kQn  k, n D 0; : : : ; nQ and nQ  1.

holds for

Now, the theorems given in Section 5.5 can be applied to construct corresponding exponential asymptotic expansions. Here, however, an additional step is needed, ."/ because the conditions are now given in terms of the intensities qij instead of the tran."/

."/

sition probabilities pij , the power moment of transition times mij Œn, and the mixed ."/

power-exponential moments ij Œ; n. Applications of the results obtained in Chapters 3–5 to an analysis of pseudo- and quasi-stationary phenomena in nonlinearly perturbed stochastic systems and nonlinearly perturbed risk processes are given in Chapters 6 and 7.

Chapter 6

Quasi-stationary phenomena in stochastic systems

Chapter 6 deals with applications of the results obtained in Chapters 3–5 to an analysis of pseudo- and quasi-stationary phenomena in nonlinearly perturbed stochastic systems. Examples of stochastic systems under consideration are queueing systems, epidemic, and population dynamics models with finite lifetimes. In queueing systems, the lifetime ."/ is usually the time at which some kind of a fatal failure occurs in the system. In epidemic models, the time of extinction of the epidemic in the population plays the role of the lifetime, while in population dynamics models, the lifetime is usually the extinction time for the corresponding population. In all examples, we study asymptotic behaviour of the probabilities Pf."/ .t / D j; ."/ > t g for the corresponding perturbed Markov/semi-Markov type process ."/ .t /, which represents the state of the system at time t , and the lifetime ."/ . We give all types of asymptotic results studied in Chapters 3–5. They include the following: (i) mixed ergodic theorems (for the state of the system) and limit theorems (for the lifetimes) that describe transition phenomena; (ii) mixed ergodic and large deviation theorems that describe pseudo- and quasi-stationary phenomena; (iii) exponential expansions in mixed ergodic and large deviation theorems; (iv) theorems on convergence of quasi-stationary distributions; and (v) asymptotic expansions for quasi-stationary distributions. In all the examples, we try to specify and to describe, in a more explicit form, conditions and algorithms for calculating the limit expressions, the characteristic roots, and the coefficients in the corresponding asymptotic expansions. As the first application, we consider a M/M queueing system that consists of N servers functioning independently of each other with exponential life and repairing ."/ ."/ times. The failure and the repairing intensities of the server i are qi; and qi;C , respectively. The state of the server i at an instant t is given by a random indicator ."/ variable i .t / that takes the value 1 if the server i works at this instant, and the value 0 if the server i is being repaired at this moment. The state of the system is described ."/ ."/ by the vector ."/ .t / D .1 .t /; : : : ; N .t //; t  0, which is, in this case, a continuous time homogeneous Markov chain. In this case, the lifetime ."/ is the first time at which the process ."/ .t / gets into the state 0N D .0; : : : ; 0/. The perturbation ."/ conditions take the form of asymptotic Taylor type expansions for the intensities qi;˙ . .0/

We consider the cases where the limiting intensities satisfy (a) qi;C > 0 for all i , while

360

6

.0/

Quasi-stationary phenomena in stochastic systems .0/

qi; D 0 for some i , or (b) qi;˙ > 0 for all i . The cases (a) and (b) correspond to pseudo- and quasi-stationary models, respectively. As the second application, we consider a M/G queueing system with one server and a bounded queue buffer of size N . We assume that the input flow is a standard Poisson flow with parameter  and the service distribution function G ."/ .t / depends on a perturbation parameter "  0. Functioning of the system is described by a random variable ."/ .t / that is the number of free places in the queue buffer at the moment t . This process is not semi-Markov but belongs to a more general class of the so-called stochastic processes with semi-Markov modulation (switchings). These processes admit a construction of imbedded semi-Markov processes. This example shows that the main results obtained in the book can be applied to more general stochastic processes, in particular, to stochastic processes with semi-Markov modulation. The lifetime ."/ is the first time at which the input item can not be placed in the queue buffer, i.e., the first time at which the process ."/ .t / hits the state N . We assume that the distribution functions G ."/ .t / weakly converge to G .0/ .t / as " ! 0 and that a Cram´er type condition holds for the service distribution function G ."/ .t / asymptotically uniformly for small ". The perturbation conditions take the form of asymptotic Taylor type expansions for power or mixed power-exponential moments the service distribution functions G ."/ .t /. We consider two cases where the limiting distribution G .0/ .t / (a) is concentrated in zero, and (b) is not concentrated in zero. The cases (a) and (b) correspond to the pseudo- and quasi-stationary models, respectively. The next application deals with birth-and-death type models described by a semiMarkov chain, where the phase space is X D f0; 1; : : : ; N g and the transition probabilities from a state i to the neighbouring states i  1 and i C 1 for 0 < i < N (N  1 ."/ ."/ ."/ and N for i D N ) have the form Qi i˙1 .t / D pi;˙ Fi .t /. The state 0 is an absorption state and the first time the process ."/ .t / hits it is interpreted as the lifetime of the corresponding stochastic system. The perturbation conditions take the form of asymp."/ totic Taylor type expansions for the transition probabilities pi;˙ and the power or mixed power-exponential moments for the distribution functions of the sojourn times ."/ .0/ Fi .t /. The limit transition probabilities satisfy the condition (a) pi;C > 0 for all .0/

.0/

i ¤ 0, while pi; D 0 for some i ¤ 0, or (b) pi;˙ > 0 for all i ¤ 0. The cases (a) and (b) correspond to pseudo- and quasi-stationary models, respectively. A standard birth-and-death type model corresponds to the case where all the distribution functions Fi."/ .t / are exponential and, therefore, ."/ .t / is a continuous time Markov chain. In this case the transition characteristics are usually defined via the ."/ ."/ ."/ ."/ ."/ transition intensities qi;˙ and have the form pi;˙ D qi;˙ =qi and Fi .t / D 1 

."/ ."/ expfqi."/ t g, where qi."/ D qi; C qi;C . In this case, the perturbation conditions take ."/

the form of asymptotic Taylor type expansions for the intensities qi;˙ . As well known, standard birth-and-death processes can be used for a description of M/M type queueing systems, epidemic or population dynamic models. We consider several examples. Let ."/ ."/ , q ."/ D .N i /q ."/ can be used us mention two of them. The model with qi; D i q C i;C

6.1 Queueing systems with highly reliable main servers

361

to describe a M/M queueing system which consists of N servers functioning indepen."/ and q ."/ , respectively. dently of each other with failure and repairing intensities q C ."/ In this case, the process  .t / represents the number of working servers at the time t . Here, the lifetime of the system is the first moment at which the number of working ."/ ."/ ."/ ."/ servers becomes zero. The model with qi; D i q , qi;C D i.N  i /qC can be used to describe an epidemic in a population with N individuals recovering independently ."/ and infecting each other pairwise independently with intensity q ."/ . with intensity q C In this case, the process ."/ .t / represents the number of ill individuals at the time t . The time of extinction of the epidemic is the first moment at which the number of ill individuals is zero. In the last example, we consider a metapopulation system consisting of N patches each of which at any time t  0 can be either occupied or empty. The state of the ."/ ."/ metapopulation at a time t is described by the vector ."/ .t / D .1 .t /; : : : ; N .t //, ."/ t  0, where i .t / is either 1 or 0, according to whether the patch i is occupied or empty at the time t . The first time the process ."/ .t / hits the state 0N D .0; : : : ; 0/ is the extinction time for the metapopulation. We assume the following: (i) in the absence of migration, the local population inhibiting a patch i will extinct in the time interval ."/ ."/ of length t with probability pi i .t / D gi i t C o.t /; (ii) the probability that an empty patch j will be colonised in a time interval of length t by migrants originating ."/ ."/ from a patch i is pij .t / D gij t Co.t /; (iii) all interaction processes (extinction and colonisation) are independent. Under the assumptions made for the local patch dynamics, we deduce that a natural time evolution model for the process ."/ .t / is a homogeneous Markov chain with absorption. The perturbation conditions take the ."/ form of asymptotic Taylor type expansions for the intensities gij . We consider the .0/

.0/

cases where the limit intensities gij > 0, for i ¤ j , while (a) gi i D 0 for some i , .0/

or (b) gij > 0 for all i; j . The cases (a) and (b) correspond, respectively, to pseudoand quasi-stationary models. Chapter 6 gives typical examples of applying results obtained from an analysis of quasi-stationary phenomena for regenerative and semi-Markov processes to a study of quasi-stationary phenomena in nonlinearly perturbed stochastic systems. In Section 6.1, we consider queueing systems with highly reliable main servers. In Sections 6.2 and 6.3, we consider M/G queueing system with one server and a bounded queue buffer. In Section 6.4, semi-Markov and Markov birth-and-death type processes are considered. Some classical models of M/M type queueing systems, epidemic or population dynamic models can be described with the use of such processes. In Section 6.5, an example of metapopulation model is considered.

6.1 Queueing systems with highly reliable main servers In this section, we study conditions on pseudo- and quasi-stationary asymptotics for queueing systems with asymptotically highly reliable servers.

362

6

Quasi-stationary phenomena in stochastic systems

6.1.1 Queueing systems with highly reliable main servers As our first application, we will consider a queueing system which consists of m servers functioning independently of each other with exponential life and repairing ."/ ."/ and qi;C , respectimes. The failure and repairing intensities of the servers are qi; tively. ."/ The state of a server i at an instant t is given by a random indicator variable i .t / that takes the value 1 if the server i is functioning at this instant, and the value 0 if the server i is being repaired at this moment. ."/ The state of the system is described by the vector ."/ .t / D .."/ 1 .t /; : : : ; m .t //, ."/ t  0. The phase space of the stochastic process  .t / is the m-dimensional binary hypercube X = fxN D .x1 ; : : : ; xm / W xi D 0; 1; i D 1; : : : ; mg. We assume that the continuous-time process ."/ .t / is a homogeneous Markov chain, continuous from the right, with the transition intensities 8 ."/ ˆ < qi; if xi D 1, yi D 0, xj D yj , j ¤ i , ."/ ."/ qxN yN D qi;C (6.1.1) if xi D 0, yi D 1, xj D yj , j ¤ i , ˆ : 0 otherwise. In this case, the parameter of the corresponding exponential distribution of the sojourn time in a state xN D .x1 ; : : : ; xm / 2 X is given by the following formula: X ."/ X ."/ X ."/ ."/ qxN yN D qi; C qi;C : (6.1.2) qxN D y¤ N xN

iWxi D1

iWxi D0

Let ."/ .n/, n D 0; 1; : : :, be subsequent moments of jumps of the Markov chain ."/ ."/ .t /, t  0, and n D ."/ . ."/ .n//, n D 0; 1; : : :, be the states of the Markov ."/ chain  .t / at subsequent moments of jumps. By the definition, the random sequence ."/ n , n D 0; 1; : : : is an imbedded discrete time Markov chain for the continuous time Markov chain ."/ .t /, t  0. Let us also define the hitting times in a state yN 2 X for the imbedded Markov ."/ ."/ ."/ chain n and the Markov chain ."/ .t / to be yN D min.n  1 W n D y/ N and ."/

."/

yN D ."/ .yN /, respectively. ."/

."/

The first hitting time 0N is the lifetime of the system. By the definition, 0N is the first time when all servers occur in a failure state. The model described above is a particular case of the model considered in Section 5.6. The notations used for the states of the Markov chain above and for the ones in Section 5.6 can easily be agreed by a natural re-numeration of states, .0; 0; : : : ; 0/ $ 0; .0; 0; : : : ; 1/ $ 1; : : : ; .1; 1; : : : ; 1/ $ 2m  1. Note that the number of nonabsorption states in this model is N D 2m  1. We will use all the notations introduced in Chapter 4 for the transition and hitting probabilities and the moment generation functions with the only change in the nota-

6.1 Queueing systems with highly reliable main servers

363

tions of indices used for the states. Instead of the indices i; j; : : : we shall use the indices x; N y; N : : :, etc. As was pointed out in Subsection 4.3.1, we need not assume that 0N D .0; : : : ; 0/ is ."/ an absorption state, since the probabilities PxN f."/ .t / D y; N 0N > t g do not depend N on the probabilities of transition from the state 0. Let us therefore see what form will the corresponding conditions formulated in Chapters 4 and 5 take. It is natural to re-formulate these conditions in terms of the ."/ intensities qi;˙ , i D 1; : : : ; m. We assume the following condition, which would allow to interpret the models with " > 0 as perturbed versions of the model with " D 0: ."/ .0/ ! qi;˙ as " ! 0, i D 1; : : : ; m. T4 : qi;˙

As far as the limit model is concerned, we assume that the following condition holds: .0/

E4 : qi;C > 0, i D 1; : : : ; m. Condition E4 can be realised in two variants. Condition T4 obviously implies that .0/ the limiting intensities satisfy qi;  0, i D 1; : : : ; m. Let us introduce the set .0/ D 0g: E D f1  i  m W qi;

The following condition is the first variant for realisation of condition E4 : .0/ > 0, i D 1; : : : ; m, and E ¤ ¿. E04 : qi;C

In this case we assume, without loss of generality, that there exists 1  l  m such that E D f1; : : : ; lg. We shall refer to servers with the indices i 2 E as “main” servers. It is obvious that conditions T4 and E04 imply that the following condition holds: ."/

."/

A7 : qi;C > 0; i D 1; : : : ; m, and qi; > 0; i D l C 1; : : : ; m, "  "1 for some "1 > 0. Condition A7 means that, for every model with the parameter " small enough, every server with index i D 1; : : : ; m can be repaired, i.e., changed from the failure state 0 to ."/ , and that every server the functioning state 1 with a positive reparation intensity qi;C with index i D l C 1; : : : ; m can break, i.e., change from the functioning state 1 to the ."/ failure state 0 with a positive failure intensity qi; . As far as servers with the indices i D 1; : : : ; l are concerned, conditions T4 and E04 imply that the corresponding failure intensities are small for " small enough and that these servers can not break, i.e., they are absolutely reliable for the unperturbed model with " D 0. This case can be interpreted as a model with asymptotically reliable main servers. As was mentioned above, conditions T4 and E04 do not guarantee that the failure ."/ intensities satisfy qi; D 0, i D 1; : : : ; l, for " small enough. In order to guarantee

364

6

Quasi-stationary phenomena in stochastic systems

this property, one should impose, additionally to T4 and E04 , the following condition: ."/

A8 : qi; D 0, i 2 E, "  "2 for some "2 > 0. Conditions A7 and A8 mean that, for every model with the parameter " that is small enough, every server with the index i D 1; : : : ; m can be repaired, every server with the index i D 1; : : : ; l can not break, i.e., it is absolutely reliable, while every server with the index i D l C 1; : : : ; m may break. The following condition is the second variant for realisation of condition E4 : .0/

E004 : qi;C > 0, i D 1; : : : ; m, and E D ¿. It is obvious that conditions T4 and E004 imply that the following condition holds: ."/

A9 : qi;˙ > 0, i D 1; : : : ; m, "  "3 for some "3 > 0. Condition A9 means that, for every model with the parameter " small enough, every server can break, i.e., to change from the functioning state 1 to the failure state 0 and can be repaired, i.e., changed from the failure state 0 to the functioning state 1. Let us note that only the first case, where condition E04 holds, actually corresponds to queueing systems with asymptotically high reliable servers. Indeed, in this case the ."/ failure intensities satisfy qi; ! 0 as " ! 0 for i 2 E. However, condition E004 can also be relevant to an analysis of queueing systems with highly reliable servers, if one would like to get the corresponding asymptotical results in the case where the failure ."/ intensities qi; , i 2 E, actually take some small values but are separated from zero as " ! 0. .0/ N where 1N D .1; 1; : : : ; 1/. Condition E4 guarantees that qxN > 0 for every xN ¤ 1, P .0/ .0/ The state 1N has a special status. For this state, we have that q1N D m iD1 qi; . If all .0/

the intensities satisfy qi; D 0, i D 1; : : : ; m, i.e., all the servers for the unperturbed .0/ system are absolutely reliable, then q1N D 0 and, therefore, the state 1N is an absorption state. Not to deal with this situation we shall use a regularisation transformation by N adding an additional intensity q > 0 to this state for the virtual transition 1N ! 1. ."/ ."/ ."/ Thus, let Q .t / D .Q 1 .t /; : : : ; Q m .t //, t  0, be, for every "  0, a semi-Markov process with a phase space X and the transition probabilities ."/

QxN yN .t / D where

."/

qQxN yN

8 ."/ ˆ qi; ˆ ˆ ˆ t g N and t  0. coincide for any x; N yN ¤ 0, Thus, we can always consider a regularised semi-Markov process Q ."/ .t / instead of the Markov chain ."/ .t /. To simplify the notations we interpret Q ."/ .t / as a regularised version of the Markov chain ."/ .t / and use the initial notation ."/ .t /; t  0, for ."/ ."/ the regularised version as well as the notations qxN yN and qxN for the corresponding transition intensities.

6.1.2 Conditions for mixed ergodic and limit/large deviation theorems Let us study relations between conditions T4 , E4 and the conditions introduced in Chapter 4 for mixed ergodic and limit/large deviation theorems. Denote N X0 D fxN 2 X W xN ¤ 0g; X1.0/ D fxN 2 X0 W x1 ; : : : ; xl D 1g; .0/

X2

.0/

D X0 n X1 ;

(6.1.6)

and U0 D X1.0/ ;

Ur D fxN 2 X2.0/ W

l X

ı.xi ; 0/ D rg;

r D 1; : : : ; l:

(6.1.7)

iD1

By the definition, l [

Ur D X2.0/ :

r D1

As was mentioned above, conditions T4 , E04 imply that condition A7 holds. According to this condition, for every 0  "  "1 , the hitting probabilities are positive, ."/ .0/ N 2 X0 ; yN 2 X1 . 0N fxN yN > 0, for any x Indeed, in this case, all reparation and failure intensities that can cause a transition from a state xN to a state yN without visiting the state 0N are positive. The property of hitting probabilities described above means that the Markov chain ."/ .0/ ."/ .t / has one class of recurrent-without-absorption states X1 X1 .

366

6

Quasi-stationary phenomena in stochastic systems

With regard to the absorption probabilities, condition A7 guarantees only that, for > 0 for any xN 2 Ul , every 0  "  "1 , the absorption probabilities satisfy yN fxN."/ 0N

yN 2 X1.0/ . Indeed, in this P case, the class Ul consists of states of the form .0; : : : ; 0; xlC1 ; : : : ; xm / such that m kDlC1 ı.xk ; 1/  1. All reparation and failure intensities, which can provide any transition within the class Ul , are positive. The one-step transition to N the Pmabsorption state 0 D .0; : : : ; 0/ is possible from the states in this class satisfying kDlC1 ı.xk ; 1/ D 1. Also, in the prelimit cases, i.e., if 0 < "  "1 , the absorption probabilities satisfy ."/ .0/ N 2 X0 n Ul , yN 2 X1 such that xi D 1 if the corresponding yN fxN 0N > 0 for the states x ."/

failure intensity are qi; > 0, and xi D 0 if the corresponding failure intensity are ."/

zero, qi; D 0. Indeed, in this case, it is possible that a transition to some state zN from the class Ul .0/ occurs without visiting the class X1 , and then a transition from the state zN to the state 0N occurs without leaving the class Ul . The situation is different in the limiting case where " D 0. In this case, as was N zN 2 X1.0/ , yN 2 X2.0/ . pointed out above, zN fxN.0/ yN D 0 for any x; This is true because all the failure intensities equal zero for the servers with the indices i 2 E. .0/ Thus, in this case, X1 is a closed set of recurrent-without absorption states, while .0/ X2 is a class of non-recurrent-without absorption states. D 0 for any xN 2 X0 n Ul , yN 2 X1.0/ , but The absorption probabilities are yN fxN.0/ 0N

.0/ yN fxN 0N

.0/

2 .0; 1/ for xN 2 Ul , yN 2 X1 . This is so also because all the failure intensities equal zero for the servers with the indices i 2 E. Thus, the case where conditions T4 and E04 hold corresponds to a pseudo-stationary .0/ model. Moreover, in this case, the class of recurrent-without-absorption states is X1 .0/ and the class of non-recurrent-without-absorption states is X2 . As was mentioned above, conditions T4 and E004 imply that condition A9 holds. According to this condition, for every 0  "  "3 , the hitting probabilities are positive, ."/ N yN 2 X0 . This means that the Markov chain ."/ .t / has one 0N fxN yN > 0, for any x;

class of recurrent-without-absorption states, X1."/ D X0 , Moreover, in this case, it can readily be seen that also all the absorption probabilities are positive, yN fxN."/ > 0, for 0N any x; N yN 2 X0 . This is true, since the reparation and failure intensities are positive for all servers. Thus, the case where conditions T4 and E004 hold corresponds to a quasi-stationary model. Moreover, in this case, the class of recurrent-without-absorption states is .0/ X1 D X0 .

6.1 Queueing systems with highly reliable main servers

367

Let us discuss how the conditions introduced in Chapters 4 are realised for the model described above. We assume that conditions T4 and E4 hold. Relations (6.1.1), (6.1.2), and condition T4 obviously imply that, for x; N yN 2 X, ."/

.0/

."/

.0/

qxN yN ! qxN yN as " ! 0;

(6.1.8)

and qxN ! qxN

as " ! 0:

(6.1.9)

By formula (6.1.3), the distributions of sojourn times are exponential and, for x; N yN 2 X, we have ."/

."/

."/

FxN yN .t / D FxN .t / D 1  expfqxN t g;

t  0:

(6.1.10)

Relations (6.1.3) and (6.1.9), and formula (6.1.10) imply that condition T3 holds. Thus, condition T1 holds. Formulas (6.1.5) and (6.1.10) imply that condition I3 holds. Formula (6.1.3) and relation (6.1.9) also imply that conditions M11 and M12 are satisfied. The non-arithmetic conditions N1 and N2 for return times to the states xN 2 X1.0/ obviously hold since the distributions of sojourn times are exponential. Let us define .0/

N  D min.qxN ; xN ¤ 0/:

(6.1.11)

By formula (6.1.5),  > 0. As was mentioned in Subsection 5.6.3, condition C12 holds for any 0 < ı <  . Conditions T4 and E04 correspond to the pseudo-stationary case. In this case, as follows from Lemma 4.5.8, condition C13 also holds. Conditions T4 and E004 correspond to the quasi-stationary case. In this case, condition G1 holds with the parameter  given in formula (6.1.11). Condition G2 also .0/ holds since the set X2 D ¿. Therefore, as follows from Lemma 4.5.10, condition C13 also holds. Conditions C14 and C15 also hold as follows from the remarks made in Section 5.6. .0/ Condition C16 is not needed, since X2 D ¿ in the quasi-stationary case. So, all the conditions introduced in Chapter 4 are verified. All results given in Chap."/ ."/ ter 4 can also be applied to the Markov chain ."/ .t / D .1 .t /; : : : ; m .t //, t  0 that describes functioning of queueing systems with highly reliable main servers.

6.1.3 Mixed ergodic and limit/large deviation theorems Conditions T4 and E04 correspond to the pseudo-stationary model with the absorption probabilities asymptotically vanishing to zero. Under these conditions, we have analogues for mixed ergodic and limit Theorems 4.4.1 and 4.4.2 and mixed ergodic and

368

6

Quasi-stationary phenomena in stochastic systems

large deviation Theorems 4.6.1 and 4.6.2, covering the case of the pseudo-stationary asymptotics. Also, analogues of Theorems 4.4.3 and 4.4.5 give conditions for convergence of the corresponding stationary distributions under the additional non-absorption condition A8 . Conditions T4 and E004 correspond to a quasi-stationary model with absorption probabilities asymptotically separated from zero. Under these conditions, we can obtain analogues of mixed ergodic and large deviation Theorems 4.6.3, 4.6.5, 4.6.7, and 4.6.11, covering the case quasi-stationary asymptotics, as well as Theorem 4.6.6 that gives conditions for convergence of the corresponding quasi-stationary distributions. It is also useful to note that the theorems listed above can be applied in the pseudostationary case as well. The limit parameters, including the stationary and quasi-stationary distributions, can be calculated by using the corresponding algorithms and formulas given in Chapter 4. They can be used for the semi-Markov processes ."/ .t /, t  0, introduced in the current section.

6.1.4 Conditions for exponential expansions in mixed ergodic and large deviation theorems Let us formulate conditions that replace the corresponding perturbation conditions introduced in Chapter 5 for the corresponding theorems representing the exponential asymptotic expansions in mixed ergodic and limit/large deviation theorems. .k/ The perturbation condition P30 can be replaced with the following condition: .k/

."/

.0/

P31 : qi;˙ D qi;˙ C qi;˙ Œ1" C    C qi;˙ Œk"k C o."k /, for i D 1; : : : ; m, where jqi;˙ Œrj < 1, r D 0; : : : ; k, i D 1; : : : ; m. .0/

It is also convenient to define qi;˙ Œ0 D qi;˙ , i D 1; : : : ; m. .k/

Note that condition P31 obviously implies that condition T4 holds. .k/ .k/ also implies that condition P30 and the corresponding expansions Condition P31 ."/ ."/ N yN 2 X: for transition intensities qxN yN and qxN take the following form for xN ¤ 0, ."/

.0/

qxN yN D qxN yN C qxN yN Œ1" C    C qxN yN Œk"k C o."k /; where, for r D 1; : : : ; k, 8 N ˆ < qi; Œr if xN ¤ 1, xi D 1, yi D 0, xj D yj , j ¤ i , qxN yN Œr D qi;C Œr if xi D 0, yi D 1, xj D yj , j ¤ i , ˆ : 0 otherwise,

(6.1.12)

(6.1.13)

and .0/ k k qx."/ N D qxN C qxN Œ1" C    C qxN Œk" C o." /;

(6.1.14)

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6.1 Queueing systems with highly reliable main servers

where, for r D 1; : : : ; k,

X

qxN Œr D

X

qi; Œr C

iWxi D1

(6.1.15)

qi;C Œr:

iWxi D0

.0/ .0/ N yN 2 X. It is also convenient to define qxN yN Œ0 D qxN yN and qxN Œ0 D qxN for xN ¤ 0, ."/

."/

As soon as the asymptotic expansions for transition intensities qxN yN and qxN are obtained, one can get, for every xN 2 X0 and n D 0; 1; : : :, asymptotic expansions for power moments Z 1 ."/ ."/ ."/ n m."/ Œn D m Œn D s n qx."/ < 1; (6.1.16) xN yN xN N expfqxN sg ds D nŠ.qxN / 0

and moment generating functions ."/ xN yN Œ; n

D D

."/ xN Œ; n

Z D

."/ nŠqxN ."/ .qxN  /nC1

1 0

."/

."/

s n e s qxN expfqxN sg ds

< 1;

 < qx."/ N :

(6.1.17)

These expansions can be obtained with the use of Lemmas 5.6.3 and 5.6.5. Note .k/ that here the only condition P31 is required. .k/ .k/ .k/ Therefore, condition P31 implies that the perturbation conditions P18 and P20 hold .;k/ .;k/  and that the perturbation conditions P24 and P26 hold for  <  . The only conditions that do require an additional analysis are the conditions O3.h/ , .h/ O4 . Recall that these conditions determine pivotal properties of asymptotic expansions ."/ .0/ for the characteristic root ."/ of the equation 0N xN xN ./ D 1 for xN 2 X1 in the most interesting pseudo-stationary case. Let us first consider the case where the following condition, additional to conditions .k/ E04 and P31 , holds: O9 : qi; Œ1 > 0, i 2 E. Condition O9 means that the failure intensities for the main (asymptotically reliable) servers are of order O."/. Let us first clarify the pivotal properties for cyclic potential quantities, ."/

."/ xN uxN yN

."/

xN ^0N

D ExN

X

."/

ı.n1 ; y/; N

.0/

xN 2 X1 ;

yN 2 X0 :

nD1

The following two cases should be considered: (1) k  l and (2) k < l. Recall .k/ that k  0 is the order of the asymptotic expansion in the perturbation condition P31 , while l  1 is the number of indices in the set E, i.e., the number of main servers.

370

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Quasi-stationary phenomena in stochastic systems ."/

As shown in Lemma 5.1.1, the potential quantities xN uxN yN admit .0; k/-expansions, ."/ xN uxN yN

D xN uxN yN Œ0 C xN uxN yN Œ1" C    C xN uxN yN Œk"k C o."k /:

(6.1.18)

.k/

Let us show that (a) in case (1), where k  l, conditions E04 , P31 , and O9 imply that .0/ the coefficients satisfy xN uxN yN Œn D 0, n < r, while xN uxN yN Œr > 0 for xN 2 X1 D Z0 , yN 2 Ur , 0  r  l (the sets Ur were defined in relations (6.1.6) and (6.1.7)). According to Lemma 5.1.2, statement (a) is equivalent to that (b) the global asymptotic order satisfies rO .x; N y/ N D r for the states xN 2 U0 ; yN 2 Ur , 0  r  l. Statement (b) follows from an obvious observation that (c) the local asymptotic order r.xN 0 ; xN 00 / equals 1 if xN 0 2 Un , xN 00 2 UnC1 for some n D 0; : : : ; l  1, or 0 otherwise. Let .xN 0 ; : : : ; xN q / be an irreducible .xN 0 ; xN 00 /-chain of states from X0 , where xN 0 D xN 0 2 U0 , xN 00 D xN q 2 Ur . Obviously, (d) the minimal number of pairs .xN n ; xN nC1 / with the local asymptotic order r.xN n ; xN nC1 / D 1 in such an irreducible .xN 0 ; xN 00 /-chain is r. These pairs correspond to transitions from some state in the class Un to some state in the class UnC1 for n D 0; : : : ; r  1. This implies statement (b) and, as a consequence, statement (a). Note that (a) implies that (e) Zr D Ur , r D 0; : : : ; l (the sets Zr were defined in S Subsection 5.1). Also, Zr D ¿, r D l C 1; : : : ; k C 1, since lnD0 Un D X0 . 0 the set of states xN D .x1 ; : : : ; xm / such that xi D 1 for some Denote by Ul1 1  i  l and xj D 0 for j ¤ i , and by Ul00 the set of states xN D .x1 ; : : : ; xm / such that xi D 1 for some l C 1  i  m and xj D 0 for j ¤ i . By the definition, 0 Ul1

Ul1 and Ul00 Ul . Formula (6.1.3) and conditions E04 and O9 imply that (f) the transition intensities S ."/ ."/ satisfy qxN 0N D 0 if xN 2 l2 nD0 Un ; (g) the transition intensities satisfy qxN 0N  qi; Œ1" 0 , for i D 1; : : : ; l, and q ."/ D 0 if as " ! 0 (note that qi; Œ1 > 0) if xi D 1, xN 2 Ul1 xN 0N ."/

.0/

0 xN 2 Ul1 n Ul1 ; (h) the transition intensities satisfy qxN 0N  qi; as " ! 0 (note that .0/

."/

qi; > 0) if xi D 1, xN 2 Ul00 , for i D l C 1; : : : ; m, and qxN 0N D 0 if xN 2 Ul n Ul00 . Using relations (f)–(h), asymptotic expansions (6.1.12) and (6.1.14) and the quotient rule for the asymptotic expansions given in statement (iv) of Lemma 8.1.1, one can readily get properties similar to (f)–(h) for the one-step absorption transition prob."/ ."/ ."/ abilities pxN 0N D qxN 0N =qxN . These properties are the following: (i) the transition probS D 0 if xN 2 l2 abilities are zero, px."/ nD0 Un ; (j) the transition probabilities satisfy N 0N ."/

.0/

."/

0 pxN 0N  qi; Œ1"=qxN as " ! 0 if xi D 1, xN 2 Zl1 for i D 1; : : : ; l, and pxN 0N D 0 if ."/

.0/

.0/

0 xN 2 Ul1 n Ul1 ; (k) the transition probabilities satisfy pxN 0N  qi; =qxN as " ! 0 if ."/

xi D 1, xN 2 Zl00 , for i D l C 1; : : : ; m, and pxN 0N D 0 if xN 2 Ul n Ul00 . Statements (i)–(k) imply that (l) the set Y0;r N (defined in Subsection 5.1.5) coincides 0 0 with Ul00 for r D 0, with Ul1 for r D 1, ¿ for r D 2; : : : ; k, and with X0 n .Ul1 [ 00 Ul / for r D k C 1.

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6.1 Queueing systems with highly reliable main servers

Statements (a) and (l) imply in an obvious way that (m) the set W0;r (defined in N 0 [ Ul00 for r D l; ¿ for Subsection 5.1.5) coincides with ¿ for r D 0; : : : ; l  1; Ul1 0 r D l C 1; : : : ; k; and X0 n .Ul1 [ Ul00 / for r D k C 1. .k/

Therefore, (n) if conditions E04 and P31 , and O9 hold and the additional assumption .l/ (1) k  l is satisfied, then condition O4 holds. In this case, by Lemmas 5.1.5 and 5.1.7, the cyclic absorption probabilities xN fxN."/ , 0N .0/ xN 2 X , have .l; k/-asymptotic pivotal expansions (where xN bN N Œl > 0), xN 0

1

."/ xN fxN 0N

D xN bNxN 0N Œl"l C    C xN bNxN 0N Œk"k C o."k /:

(6.1.19) ."/

Also, by Theorem 5.4.1, the characteristic root ."/ of the equation 0N xN xN ./ D 1 has the following .l; k/-asymptotic pivotal expansion: ."/ D al "l C    C ak "k C o."k /:

(6.1.20) .k/

Similarly it can be shown that (o) in case (2) with k < l, conditions E04 , P31 , and O9 imply that xN uxN yN Œn D 0, n < r, while xN uxN yN Œr > 0 for xN 2 X1.0/ D U0 , yN 2 Ur ; 0  r  k (these sets were defined in relations (6.1.6) and (6.1.7)), and (p) S .0/ N 2 X1 D U0 , yN 2 lrDkC1 Ur . xN uxN yN Œn D 0, n < k C 1, for x The proposition (o) implies that (q) Zr D Ur , r D 0; : : : ; k, and ZkC1 D Sl N were defined in statement (l). r DkC1 Ur . The sets Y0;r coincides with ¿ for It follows from statements (o) and (l) that (r) the set W0;r N D X . r D 0; : : : ; k and W0;kC1 0 N .k/

Therefore, (s) if conditions E04 and P31 , and O9 hold and the additional assumption .k/ (1) k  l is satisfied, then condition O3 holds. ."/ .0/ By Lemmas 5.1.5 and 5.1.7, the cyclic absorption probabilities xN fxN 0N , xN 2 X1 ."/

.0/

satisfy the following relations: (t) xN fxN 0N D o."k /, xN 2 X1 . Also, by Theorem

5.4.1, the characteristic root ."/ of the equation 0N x."/ N xN ./ D 1 satisfies the relation (u) ."/ D o."k /. The asymptotic relations and (t) and (u) are not surprising. One should not expect that the lemmas and theorem mentioned above would give more precise asymptotics under the perturbation conditions providing exact asymptotics for transition intensities only up to terms of order "k . Let us also consider a more complicated case where the following condition, addi.k/ tional to conditions E04 and P31 , holds: O10 : qi; Œn D 0, n < ki and qi; Œki  > 0 for some 1  ki  k, for every i 2 E. Condition O10 means that the failure intensity for main (asymptotically reliable) servers i 2 E are of a different orders O."ki /.

372

6

Quasi-stationary phenomena in stochastic systems

As above, two cases should be considered, (3) k  k1 C    C kl D kNl and (4) k < kNl . By repeating the asymptotic analysis presented above, it can be shown that, under .k/ .kN / .k/ conditions E04 , P31 , and O10 , condition O3 l holds in case (3), or condition O4 holds .;h/ .;h/ in case (4). Conditions O5 , O6 , as was mentioned in Subsection 5.3.7, can not be expressed so explicitly in terms of coefficients in the asymptotic expansions .k/ used in the perturbation condition P31 . Here, we need to use the coefficients of the .0/ corresponding expansions for the moment generating functions 0N x."/ N xN ./, x 2 X1 , given in Lemma 5.3.5 and specified for the semi-Markov processes considered in the current section.

6.1.5 Exponential expansions in mixed ergodic and large deviation theorems All results given in Chapter 5 can be specified for semi-Markov processes ."/ .t /, t  0 that describe functioning of queueing systems with highly reliable main servers. .k/ Conditions T4 , E04 , P31 , and O9 or O10 correspond to a pseudo-stationary model with the absorption probabilities asymptotically vanishing to zero. Under these conditions, we can formulate analogues of Theorems 5.4.1 and 5.4.2, which would give exponential asymptotic expansion in mixed ergodic and large deviation theorems. Also an analogue of Theorem 5.5.1 gives higher order asymptotic expansions for stationary distributions under the additional non-absorption condition A8 . .k/ correspond to a quasi-stationary model with the abConditions T4 , E004 , and P31 sorption probabilities asymptotically separated from zero. Under these conditions, analogues of Theorems 5.4.3 and 5.4.4, which give exponential asymptotic expansion in mixed ergodic and large deviation theorems, can be obtained. Also, the analogues of Theorems 5.5.4 and 5.5.5 give higher order asymptotic expansions for the quasistationary distributions. It is useful to note that the theorems given above also cover .k/ the pseudo-stationary case, i.e., they can be applied if conditions T4 , E04 , and P31 hold as well. The limit parameters, including the stationary and quasi-stationary distributions, and the coefficients in the corresponding asymptotic expansions, can be calculated by using the corresponding algorithms and formulas given in Chapters 4 and 5. They should be specified for the semi-Markov processes ."/ .t /, t  0, introduced in the current section.

6.2 M/G queueing systems with quick service In this section, we investigate conditions for quasi- and pseudo-stationary asymptotics for M/G queueing systems with quick service and a bounded queue buffer. In this case, the perturbed stochastic processes, which describe the dynamics of the queue in the

6.2

M/G queueing systems with quick service

373

system, belong to the class of so-called stochastic processes with semi-Markov modulation. They admit a construction of imbedded semi-Markov processes and are more general than semi-Markov processes. An example presented in this section shows how the main results obtained in the book can be applied to stochastic processes more general than semi-Markov processes, in particular, to stochastic processes with semiMarkov modulation.

6.2.1 M/G queueing systems with quick service Let us consider a standard M/G queueing system with a bounded queue buffer. This system has one server and a bounded queue buffer of size N  2 (including the place in the server). We make the following assumptions: (a) the input flow of customers is a standard Poisson flow with parameter  > 0; (b) the customers are placed in the queue buffer in the order they come in the system; (c) a new customer coming in the system is immediately served if the queue buffer is empty, is placed in the queue buffer if the number of customers in the system is less than N , or leaves the system if the queue buffer is full, i.e., the number of customers in the system equals to N ; (d) the service times for different customers are independent random variables with the distribution function G ."/ .t /; (f) the processes of a customer coming in the system and the service processes are independent. We assume that the distribution of the service time depends on the perturbation parameter "  0 and that G ."/ .t / is a distribution concentrated on the interval Œ0; 1/ and is not concentrated in zero for every " > 0. The state of the system is determined by a random variable ."/ .t /, which is the number of empty places in the queue buffer at the moment t . The stochastic process ."/ .t /, t  0 has a space of states, X D f0; 1; : : : ; N g. We assume that the process ."/ .t /, t  0 is right continuous. We also assume that the initial queue is ."/ .0/ D const > 0. The lifetime of the system ."/ D inf.t > 0 W ."/ .t / D 0/ is defined as the first moment when the number of empty places in the queue buffer becomes equal 0. Instead of the stochastic process ."/ .t /, t  0, one can also use the process Q ."/ .t / D N  ."/ .t /, t  0. Obviously, the random variable Q ."/ .t / is the number of customers in the queue buffer at the moment t . In this case, the lifetime of the system ."/ D inf.t > 0 W Q ."/ .t / D N / is defined to be the first moment at which the number of customers in the queue buffer becomes equal N . Note that the definition of the lifetime given above agrees with the conventions used in Chapters 4 and 5, where the phase space for the corresponding processes is always the set X D f1; : : : ; N g and the absorption times are always the first hitting time in the state 0. In the example considered in the current section, the lifetime ."/ , in fact, can be interpreted as the first time at which the customer, coming in the system with the actual queue buffer size N  1, leaves the system because the number of customers in the system is N  1, i.e., the queue buffer is full.

374

6

Quasi-stationary phenomena in stochastic systems

We assume the following condition, which allows us to interpret the models with " > 0 as perturbed versions of the model with " D 0, T5 : G ."/ ./ ) G .0/ ./ as " ! 0, where G .0/ .t / is a proper distribution function concentrated on the interval Œ0; 1/. There are two cases possible. The first one corresponds to a model with asymptotically quick service subject to the following condition: E05 : G .0/ .t / is a distribution function concentrated in 0, i.e., G .0/ .0/ D 1. This is the case with pseudo-stationary asymptotics for the corresponding probabilities conditioned by non-absorption events. For the second case, which corresponds to a model with the usual service, the following condition holds: E005 : G .0/ .t / is a distribution function not concentrated in 0, i.e., G .0/ .0/ < 1. This is the case for quasi-stationary asymptotics of the corresponding probabilities conditioned by non-absorption events.

6.2.2 An imbedded semi-Markov process The process ."/ .t /, t  0 is a regenerative process. The subsequent moments at which this process hits the state N , which results in the end of the service for some customer, ."/

n , n D 1; 2 : : :, can be taken as subsequent regeneration moments for this process. In fact, the subsequent moments at which this process hits any other fixed state i > 1 can be taken to be subsequent regeneration moments as well. Note that the state 1 can not be used to construct appropriate regeneration cycles. In this case, the corresponding stopping probability for one regeneration cycle is f ."/ D Pf."/
i C 1, if 1  i  N  1, 2  j  i C 1, if 1  i  N  1, j D 1,

(6.2.2)

if 1  i  N  1, j D 0, if i D N , j D N  1, if i D N , j ¤ N  1.

The zero value of the transition probability in the first line in the right-hand side of (6.2.2) shows that the state 0 is an absorption state. Zero in the second line reflects the impossibility of transition from the state 1  i  N  1 to the state j > i C 1, which results in the end of the service for one customer. The expression for the transition probability in the third line is Pf&1."/  t;  ."/ .t / D i  j C 1g, i.e., the probability that the time of the service for the first customer is less than or equal to t and that exactly i  j C 1 customers come in the system during this time. Zero in the fourth line reflects the impossibility of transition from the state 1  i  N  1 to the state j D 1, because the transition in the absorption state 0 occurs if i customers come in the system before the end of the service of the first customer.

376

6

Quasi-stationary phenomena in stochastic systems ."/

."/

The expression for the transition probability in the fifth line is Pf 1 C    C i  ."/ min.t; &1 /g, i.e., the probability that i customers come in the system before the end of the service of the first customer and before time t . The expressions in the sixth and seventh lines reflect the fact that the waiting time for the first customer coming in the system, after the moment when the queue buffer becomes empty, has an exponential distribution. As usual, we use the symbols Pi .A/ and Ei for the conditional probabilities and the expectations calculated under the condition that ."/ .0/ D ."/ .0/ D i . An important property of the process ."/ .t / and the imbedded semi-Markov process ."/ .t / is that the first hitting time to the state 0, denoted above by ."/ , is the same for both processes. Also the first hitting time to the state N , which results in the end of the service of a customer, denoted above by 1."/ , is the same for both processes. It follows from this observation that, for any initial state 1  i  N , the hitting ."/ ."/ probabilities 0 fiN and the absorption probabilities Nfi0 defined for the imbedded semi-Markov processes ."/ .t / coincide, respectively, with the hitting probabilities ."/ ."/ Pi f 1  ."/ g and the absorption probabilities Pi f."/  1 g defined for the ."/ process .t /. ."/

Moreover, for any initial state 1  i  N , the distribution of hitting times, 0 GiN .t /, ."/ and the distribution of absorption times, N Gi0 .t /, defined for the imbedded semi."/ Markov processes ."/ .t / coincide, respectively, with the distributions Pi f 1  t;

1."/  ."/ g and the absorption probabilities Pi f."/  t ; ."/ < 1."/ g defined for the process ."/ .t /. Let the initial state satisfy ."/ .0/ D i . In terms of the characteristics used for re."/ generative processes in Chapter 3, the distribution function 0 GN N .t / plays the role of the distribution function for the duration of the regeneration period F ."/ .t / D ."/ ."/ ."/ PN f 1  t ; 1  ."/ g; the absorption probability NfN 0 replaces the stopping probability for one regeneration cycle f ."/ D 1  F ."/ .1/; the moment generatR1 ."/ ."/ ing function 0 N N ./ D 0 e t 0 GN N .dt / is now the moment generating function ."/  ."/ ./; and the characteristic root ."/ is the root of the equation 0 N N ./ D 1. Also, ."/ .t / replaces the distribution function for the duration of the distribution function i GiN ."/ the transition period FQ .t / D Pi f 1."/  t; 1."/  ."/ g; the absorption probabil."/ ity ifN 0 is the stopping probability for the transition period fQ."/ D 1  FQ ."/ .1/; R1 ."/ ."/ the moment generating function 0 iN ./ D 0 e t 0 GiN .dt / becomes the moment ."/ generating function Q ."/ ./; and the quantity 0 iN .."/ / replaces the quantity Q ."/ . Now, we can apply the methods of asymptotic analysis for perturbed semi-Markov processes developed in Chapter 4 and 5 to asymptotic analysis of the corresponding characteristics related to regeneration cycles of the regenerative process ."/ .t /.

6.2

M/G queueing systems with quick service

377

The coincidence for the corresponding absorption probabilities and the hitting distributions for the processes ."/ .t / and ."/ .t / should not lead to a misconception of the probabilities Pi f."/ .t / D j; ."/ > t g and Pi f ."/ .t / D j; ."/ > t g, which are the main object of our asymptotic analysis. By the definition, the trajectories of the processes ."/ .t / and ."/ .t / in the inter."/ val Œ0; ."/  are synchronised at the moments ˛n ; n   ."/ . But trajectories of the process ."/ .t / may deviate from the corresponding trajectories of the process ."/ .t / between these moments due to new customers coming in the system. That is why the probabilities Pi f."/ .t / D j; ."/ > t g and Pi f ."/ .t / D j; ."/ > t g may not coincide, as well as the corresponding quasi-stationary distributions and other related quasi-stationary quantities. The process ."/ .t /, t  0, belongs to the class of so-called stochastic processes with semi-Markov modulation (switchings). We refer to books by Gikhman and Skorokhod (1971), Silvestrov (1980a), Koroliuk and Limnios (2005), and Anisimov (2008), where one can find a description of such processes.

6.2.3 Hitting and absorption probabilities for the imbedded semi-Markov process Let us study relations between the conditions T5 , E05 , E005 , and the conditions for the mixed ergodic and limit/large deviation theorems introduced in Chapters 3 and 4 for the imbedded semi-Markov process ."/ .t /, t  0. It follows from formula (6.2.2) that the transition probabilities for the imbedded ."/ discrete time Markov chain n , n D 0; 1; : : :, corresponding to the semi-Markov process ."/ .t /, t  0, has the following form:

."/ pij

8 ˆ 0 ˆ ˆ ˆ ˆ 0 ˆ ˆ ˆ ˆ ."/ ˆ ˆ < pij C1 ./ D 0 ˆ ˆ ˆ ˆ pNi."/ ./ ˆ ˆ ˆ ˆ ˆ 1 ˆ ˆ : 0

if i D 0, if 1  i  N  1, j > i C 1, if 1  i  N  1, 2  j  i C 1, if 1  i  N  1, j D 1,

(6.2.3)

if 1  i  N  1, j D 0, if i D N , j D N  1, if i D N , j ¤ N  1,

where pr."/

D

pr."/ ./

Z D

1 0

.s/r s ."/ e G .ds/; rŠ

r D 0; 1; : : : ;

(6.2.4)

378

6

Quasi-stationary phenomena in stochastic systems

and ."/ ."/ D pNrC1 ./ D pNrC1

Z

1 0

D1

r C1 s r s e .1  G ."/ .s// ds rŠ

r X

."/

pl ./;

r D 0; 1; : : : :

(6.2.5)

lD0

The corresponding matrix of transition probabilities has the following form: 2 3 1 0 0 0 ::: 0 0 6 pN ."/ 0 p ."/ 0 ::: 0 0 7 6 1 7 0 6 ."/ 7 ."/ ."/ 6 7 p N 0 p p : : : 0 0 2 1 0 P."/ D 6 (6.2.6) :: :: :: :: :: :: :: 7 6 7: 6 : : : : : : : 7 6 ."/ ."/ ."/ ."/ ."/ 7 4 pN 5 0 p p : : : p p 1 0 N 1 N 2 N 3 0 0 0 0 ::: 1 0 By the assumptions on the initial model, the distribution G ."/ .t / is assumed to be not concentrated in zero for every " > 0. This assumption implies that (a) pk."/ > P ."/ ."/ 0; k D 0; 1; : : :; (b) 1 kD0 pk D 1; (c) 0 < pNk < 1, k D 0; 1; : : : . Condition T5 implies that condition T1 (a) holds, i.e., that the following relations hold for i D 0, j 2 X: ."/

.0/

pij ! pij

as " ! 0:

(6.2.7)

Indeed, formulas (6.2.2)–(6.2.5) and condition T5 imply that the following relations holds for r D 0; : : :: pr."/ ! pr.0/ as " ! 0;

(6.2.8)

pNr."/ ! pNr.0/ as " ! 0:

(6.2.9)

and

Relation (6.2.8) follows from continuity and boundedness of the integrands in the integrals which define the probabilities pn."/ and condition T5 . Relation (6.2.9) is a direct corollary of relation (6.2.8). Relations (a)–(c) imply that the hitting probabilities satisfy (d) 0 fij."/ > 0, i ¤ 0,

j ¤ 0; 1; the hitting probabilities are zero, (e) 0 fi1."/ D 0, i ¤ 0; (f) the absorption probabilities are positive jfi0."/ > 0, i; j ¤ 0. Thus, in the prelimit case, i.e., for " > 0, the class of recurrent-without-absorption ."/ ."/ states is the set X1 D f2; : : : ; N g if N > 2 or X1 D f1; 2g if N D 2, while the

6.2

379

M/G queueing systems with quick service ."/

class of non-recurrent-without-absorption states is the set X2 D f1g if N > 2 or ."/ X2 D ¿ if N D 2. .0/ .0/ .0/ If condition E05 holds, (g) p0 D 1 and pn D 0, n D 1; 2; : : :; (h) pNn D 0, n D 1; 2; : : : . In this case, the corresponding matrix of the transition probabilities takes the following form: 2 3 1 0 0 0 ::: 0 0 6 0 0 1 0 ::: 0 0 7 6 7 6 0 0 0 1 ::: 0 0 7 6 7 P.0/ D 6 :: :: :: :: :: :: :: 7 : (6.2.10) 6 : : : : : : : 7 6 7 4 0 0 0 0 ::: 0 1 5 0 0 0 0 ::: 1 0 .0/

In this case, (i) the hitting probabilities satisfy 0 fij

D 1 for any 1  i < j  N .0/

and i D N , j D N  1; N ; (j) the hitting probabilities are zero, 0 fij

D 0, for any .0/

1  j  i < N and i D N , j < N  1; (k) the absorption probabilities are jfi0 D 0 for any i; j ¤ 0. .0/ Thus, in this limit case, i.e., for " D 0, the set X1 D fN  1; N g is the class of .0/ recurrent-without-absorption states while the set X2 D f1; : : : ; N  2g is the class of non-recurrent-without-absorption states. P .0/ If condition E005 holds then (l) pn.0/ > 0, n D 0; 1; : : :; (m) 1 nD0 pn D 1; (n) .0/ 0 < pNn < 1, n D 1; 2; : : : . In this case, the corresponding matrix of transition probabilities has the same structure as in the prelimit case. This implies that (o) 0 fij.0/ > 0, i ¤ 0, j ¤ 0; 1; (p) 0 fi1.0/ D 0, i ¤ 0; (q)

.0/ jfi0

> 0, i; j ¤ 0. Thus, in the limit case, i.e., for " D 0, we have X1.0/ D f2; : : : ; N g and X2.0/ D f1g .0/ .0/ if N > 2, or X1 D f1; 2g and X2 D ¿ if N D 2. Condition T5 also implies that condition T1 (b) holds, i.e., that the following relation holds for i ¤ 0, j 2 X: ."/

.0/

Qij ./ ) Qij ./ as " ! 0:

(6.2.11)

Relation (6.2.11) follows from continuity and boundedness of the integrands in the ."/ integrals which define the probabilities Qij .t / and condition T5 . It should be noted that in the case where condition E05 holds, the distributions of .0/ .0/ sojourn times, Fi .t /, are concentrated in 0, i.e., Fi .0/ D 1 for 1  i  N  1. Thus, these states are instant for the limit process. However, the distribution function .0/ is FN .t / D 1  e t . This directly follows from formulas (6.2.2). .0/ In the case where condition E005 holds, the distributions of sojourn times, Fi .t /, are not concentrated in zero for any i ¤ 0. This also follows from formulas (6.2.2).

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6.2.4 Asymptotics of absorption probabilities in the pseudo-stationary case Let us discuss in more details the question about the asymptotic behaviour of absorption probabilities in the pseudo-stationary case where conditions T5 and E05 hold. We say that positive functions a" ; " > 0, and b" ; " > 0, are comparable, if a" =b" ! c as " ! 0, where 0  c  1. We shall use the symbols a" b" if c D 0; a"  b" if 0  c < 1; and a"  b" if 0 < c < 1, respectively. Recall also that the notation a"  b" was used in the case c D 1. ."/ The following relations can be written for the absorption probabilities iC1 fi0 , i D 1; : : : ; N  1, ."/ iC1 fi0

D pNi."/ C

i1 X

."/ ."/ pik iC1 fkC10 :

(6.2.12)

kD1

Transitions from a state 1  i  N  1 are possible with positive probabilities only to the states j  i C 1. This obviously implies that, for any 1  i < r < j  N , ."/ 0 fij

."/

D 0 fir

."/ 0 frj :

(6.2.13)

This relation implies the following formula for any 1  i < j  N : ."/ ."/ jfi0 D iC1 fi0 C

jX 1

."/ ."/ 0 fir rC1 fr 0 :

(6.2.14)

r DiC1

Using formula (6.2.14) on can rewrite equations (6.2.12) in the following form for i D 1; : : : ; N  1: ."/ iC1 fi0

."/

D pNi

C

i1 X

."/

."/

pik .kC2 fkC10 C

kD1

i X

."/ ."/ 0 fkC1r r C1 fr 0 /:

(6.2.15)

rDkC2

This relations can be rewritten in the following form for i D 1; : : : ; N  1:   P Pi1 ."/ ."/ ."/ ."/ pNi."/ C i2 pik 0 fkC1r r C1 fr 0 kC2 fkC10 C kD1 rDkC2 ."/ : (6.2.16) iC1 fi0 D P ."/ ."/ ."/ 1  p1  i2 p f 0 kD1 ik kC1i ."/

."/

If conditions T5 and E05 hold, then (r) pn ! 0 as " ! 0 for n  1; (s) 0 fij ! 1 ."/

as " ! 0 for 1  i < j  N ; and (t) jfi0 ! 0 as " ! 0 for 1  i < j  N . Using relations (r)–(t), we get from (6.2.16) the following asymptotic relations for i D 1; : : : ; N  1: ."/

."/

iC1 fi0  pNi

C

i2 X kD1

."/

pik

i1 X r DkC1

."/ r C1 fr 0

as " ! 0:

(6.2.17)

6.2

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M/G queueing systems with quick service

By regrouping terms in the sum in the right-hand side of (6.2.17) we can rewrite this relation for i D 1; : : : ; N  1 in the following equivalent form: ."/ iC1 fi0

."/

 pNi

C

i1 X

i1 X

."/

kC1 fk0

pr."/ as " ! 0:

(6.2.18)

r DikC1

kD2

."/

."/

."/

In particular, the asymptotic relations (6.2.18) yield that 2 f10  pN1 ; 3 f20  ."/ ."/ ."/ ."/ ."/ ."/ ."/ ."/ ."/ ."/ ."/ ."/ pN2 ; 4 f30  pN3 C 3 f20 p2 ; 5 f40  pN4 C 4 f30 .p2 C p3 / C 3 f20 p3 as " ! 0, etc. The recurrence asymptotic relations (6.2.18) allow to make a desirable asymptotic analysis for absorption probabilities. Note that for the cyclic absorption probability, we have ."/ NfN 0

."/

D NfN 1 0 :

(6.2.19)

Let us introduce the following asymptotic order condition: O11 : pNk."/ pr."/ pNi."/ for 2  k  i  1, i  k C 1 < r  i  1, 2 < i  N  1. Let us show that, under conditions T5 , E05 , condition O11 is equivalent to the following asymptotic relations for absorption probabilities for j D 1; : : : ; N  1: ."/ j C1 fj 0

."/

 pNj

as " ! 0:

(6.2.20)

The asymptotic relations (6.2.18) directly imply that the asymptotic relation in (6.2.20) holds for j D 1; 2. Let us first assume that condition O11 holds and that the asymptotic relation in (6.2.20) is true for j D 1; : : : ; i  1, where 2 < i < N  1, and prove that, in this case, the asymptotic relation in (6.2.20) holds for j D i . Indeed, the induction assumption made above and condition O11 permit to transform asymptotic relation (6.2.18) for j D i to the following form: ."/ iC1 fi0



pNi."/

C

i1 X

."/ kC1 fk0

i1 X

pr."/

r DikC1

kD2

 pNi."/ C

i1 X

i1 X

pNk."/ pr."/  pNi."/ as " ! 0:

(6.2.21)

kD2 r DikC1

Assume now that the asymptotic relation (6.2.20) holds for j D 1; : : : ; N  1 and the asymptotic relation in condition O11 is verified for i D 1; : : : ; j  1, where 2 < j < N  1, and prove that the asymptotic relation in condition O11 holds for i D j.

382

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Indeed, the induction assumptions made above permit to transform the asymptotic relation (6.2.18) for i D j to the following form: pNj."/ 

."/ j C1 fj 0

 pNj."/ C

jX 1 kD2

 pNj."/ C

jX 1

jX 1

jX 1

."/ kC1 fk0

pr."/

rDj kC1

pNk."/ pr."/ as " ! 0;

(6.2.22)

kD2 r Dj kC1

and, therefore, jX 1

jX 1

."/

pNk pr."/ ! 0 as " ! 0:

(6.2.23)

kD2 r Dj kC1

Relation (6.2.23) implies that the asymptotic relation in condition O11 holds for i D j. Introduce the following asymptotic order condition: ."/

."/

O12 : pNiC1 pNi , for i D 1; : : : ; N  1. Let us show that, under conditions T5 , E05 , condition O12 is equivalent to the following condition: ."/

O012 : pNi

."/

 pi

as " ! 0, for i D 1; : : : ; N  1. ."/

Indeed, the identities pNi

."/

."/

pi =pNi

."/

."/

C pNiC1 =pNi

."/

D pi ."/

."/

 pi =pNi

."/ ."/ ."/ ."/ pi =pNi C pNiC1 =pNi

."/

C pNiC1 ; i  1, condition O12 imply that 1 D as " ! 0, while condition O012 implies that

."/ ."/ 1C pNiC1 =pNi

."/

."/

1D  as " ! 0 and, therefore, pNiC1 =pNi ! 0 as " ! 0. Finally, under conditions T5 , E05 , O11 , and O12 , we can rewrite the asymptotic relations (6.2.20) in the following simpler form for i D 1; : : : ; N  1: ."/ iC1 fi0

."/

 pi

as " ! 0:

(6.2.24)

The following condition gives a typical example where both conditions O11 and O12 obviously hold: ."/

O13 : pNi

 pNi "i as " ! 0 for i D 1; : : : ; N 1, where pNi > 0 for i D 1; : : : ; N 1.

Let us show that, under conditions T5 , E05 , condition O13 is equivalent to the following asymptotic relations for absorption probabilities for j D 1; : : : ; N  1: ."/ j C1 fj 0

 pNj "j as " ! 0;

where pNj > 0, for j D 1; : : : ; N  1.

(6.2.25)

6.2

383

M/G queueing systems with quick service

The proofs is needed only for the converse statement that the asymptotic relations (6.2.25) for j D 1; : : : ; N  1 imply condition O13 . Indeed, let us assume that the asymptotic relations (6.2.25) hold for j D 1; : : : ; N  1, and that the asymptotic relation in condition O13 holds for i D 1; : : : ; j  1, where 1  j < N  1, and prove that the asymptotic relation in condition O13 holds for i D j. ."/ ."/ The induction assumptions made above mean that pNi  pi  pNi "i as " ! 0 for i D 1; : : : ; j  1. This relations permits to transform the asymptotic relation (6.2.25) for i D j to the following form: j

pNj " 

."/ j C1 fj 0



."/ pNj

C

jX 1 kD2

."/

 pNj C

jX 1

jX 1

jX 1

."/ kC1 fk0

pr."/

rDj kC1

pNk "k pNr "r as " ! 0:

(6.2.26)

kD2 r Dj kC1

Relation (6.2.26) obviously implies that ."/

pNj

 pNj "j as " ! 0:

(6.2.27)

The following condition, which is more general than O13 , also gives a typical example in which conditions O11 and O12 obviously hold: O14 : pNi."/  pNi "hi as " ! 0 for i D 1; : : : ; N  1, where (a) pNi > 0 for i D 1; : : : ; N  1, and (b) hk C hr > hi if k C r > i for 2  k; r; i  N  1. A proof similar to the one given above yields that, under conditions T5 , E05 , condition O14 is equivalent to the following asymptotic relations for absorption probabilities for j D 1; : : : ; N  1: ."/ j C1 fj 0

 pNj "hj as " ! 0;

(6.2.28)

where pNj > 0 for j D 1; : : : ; N  1. Let us consider a model in which conditions T5 , E05 , O11 , and O12 hold. Assume that the following condition, which implies condition T5 , holds: T6 : G ."/ .t / D G.t =."/ /, where (a) G.t / is a proper distribution function concentrated on the interval Œ0; 1/ and not concentrated in zero, and (b) ."/ > 0 for " > 0 and ."/ ! .0/ < 1 as " ! 0. There are two cases to consider. The first one corresponds to a model with asymptotically quick service, where the following condition holds: E06 : .0/ D 0.

384

6

Quasi-stationary phenomena in stochastic systems

This is the case with pseudo-stationary asymptotics for the corresponding probabilities conditioned by non-absorption events. The second case, which corresponds to a model with the usual service; here the following condition holds: E006 : .0/ > 0. The model described above can be interpreted in the following way. Let us change the time scale such that the time t is counted as the time tQ D t =."/ in the new time scale. In the new time scale, the lifetime will take the value Q ."/ D ."/ =."/ . Correspondingly, the service time will have the distribution function G.t / and the input flow of customers will be a Poisson flow with the parameter ."/ . This flow has a lower intensity, if the parameter ."/ takes a small value. In particular the model can be interpreted as a queueing system with an asymptotically lower intensity of input flow if ."/ ! 0 as " ! 0. ."/ In the model described above, the probabilities pi , i D 0; 1; : : :, are given by the following formulas: Z 1 Z 1 .s/i s .s/i ."/ s ."/ ."/ ."/ i pi D e e G.ds= / D . / G.ds/: (6.2.29) iŠ iŠ 0 0 Let us consider the pseudo-stationary case, where condition E06 holds. Denote Z 1 s n G.ds/; n D 1; 2; : : : : Mn D 0

Let us also assume the following condition to hold: R1 .N/ M14 : MN D 0 s N G.ds/ < 1. Conditions T6 , E06 , and M.N/ 14 imply that the following asymptotic relations hold for i D 1; : : : ; N  1: i Mi as " ! 0: (6.2.30) iŠ Also, using a standard representation of the exponential function in the form of a power series, we get for i D 1; : : : ; N  1 that   Z 1 ."/ iC1 .."/ s/iC2 ."/ ."/ s . s/ e C C    G.ds/ pNiC1 D .i C 1/Š .i C 2/Š 0   Z 1 .."/ s/iC1 ."/ s .."/ s/ 1C D e C    G.ds/ .i C 1/Š .i C 2/ 0   Z 1 .."/ s/iC1 ."/ s .."/ s/  1C e C    G.ds/ .i C 1/Š 1Š 0 Z 1 .."/ s/iC1 iC1 MiC1 G.ds/ D .."/ /iC1 : (6.2.31) D .i C 1/Š .i C 1/Š 0 ."/

pi

 .."/ /i 

6.2

385

M/G queueing systems with quick service

Relations (6.2.30) and (6.2.31) imply in an obvious way that both conditions O11 and O12 hold. .N/ Thus, under conditions T6 , E06 , and M14 , the following asymptotic relations hold for i D 1; : : : ; N  1: ."/ iC1 fi0

."/

 pNi

."/

 pi

 .."/ /i 

i Mi as " ! 0: iŠ

(6.2.32)

.N/

Let us assume that, additionally to T6 , E06 , and M14 , the following condition holds: O15 : ."/  1 " as " ! 0, where 1 > 0. In this case, the following asymptotic relations hold for i D 1; : : : ; N  1: ."/ iC1 fi0

."/

 pNi

 .."/ /i 

i Mi .1 /i Mi i  " as " ! 0: iŠ iŠ

(6.2.33)

Thus, condition O13 holds. .N/ Let us now assume that, additionally to T6 , E06 , and M14 , the following condition holds: O16 : ."/  h "h " as " ! 0, where h > 0 and h is a positive integer number. In this case, the following asymptotic relations hold for i D 1; : : : ; N  1: .h /i Mi ih i Mi  " as " ! 0: (6.2.34) iŠ iŠ Thus, condition O14 holds with the parameters hi D hi , i D 1; : : : ; N  1. The following example shows that there are models in which the asymptotic order condition O11 holds but condition O12 does not. Let us consider a model where the distribution G ."/ .t / has two atoms in the point 0 and ."/ > 0 with the probabilities 1  q ."/ and q ."/ > 0, respectively. We assume that (v) q ."/ ! q .0/ D 0 as " ! 0, and (w) ."/ ! .0/ as " ! 0, where 0  .0/  1. The assumption (v) obviously implies that conditions T5 and E05 hold. ."/ In this example, the probabilities pi ; i D 0; 1; : : :, are given by the following formulas: .."/ /i ."/ ."/ ."/ e q : (6.2.35) pi D iŠ Let i < j C r; 1  i; j; r  N  1. The following asymptotic estimate takes place: P P1 .."//k  ."/ ."/ .."//r ."/ ."/ i1 .."/ /k ."/ ."/ C q  rŠ e q e pNj pr kDj kDi kŠ kŠ D P ."/ 1 .."/ /k ."/ ."/ pNi e q kDi kŠ ."/ iC1 fi0

 e

 pNi."/  .."/ /i 

."/ ."/

q

."/



i1 X .."/ /kCr i1 i Š kŠrŠ

kDj ."/

C e  q ."/

.."/ /r ! 0 as " ! 0: rŠ

(6.2.36)

386

6

Quasi-stationary phenomena in stochastic systems

Indeed, the expression in the right-hand side of (6.2.36) tends to 0 as " ! 0 whatever value the limit 0  .0/  1 takes. In case .0/ D 0, this is true because ."/ q ."/ ! 0 and ."/ ! 0 as " ! 0, while e  ! 1 as " ! 0. Note also that all power indices in the sum in the right-hand side of (6.2.36) are k C r  i  1  0 if j  k  i  1. If 0 < .0/ < 1, this can be seen, since q ."/ ! 0 and ."/ ! .0/ as ."/ " ! 0. In case .0/ D 1, this is true, since q ."/ ! 0 and e  .."/ /kCr ! 0 as " ! 0. Relation (6.2.36) implies that condition O11 holds and, therefore, the following asymptotic relations take place for i D 1; : : : ; N  1: ."/ iC1 fi0

 pNi."/ as " ! 0:

(6.2.37)

If .0/ D 0, then for i  1, pi."/  .."/ /i q ."/ 

i as " ! 0; iŠ

(6.2.38)

and ."/ ."/ ."/ pNiC1 D piC1 C piC2 C 

D .."/ /iC1 q ."/

."/ iC1 ."/ .1 C C    /e  .i C 1/Š i C2

 .."/ /iC1 q ."/

iC1 : .i C 1/Š

(6.2.39)

Therefore, condition O12 also holds and the following asymptotic relation takes place for i D 1; : : : ; N  1: ."/ iC1 fi0

."/

 pNi

."/

 pi

 .."/ /i q ."/ 

i as " ! 0: iŠ

(6.2.40)

If 0 < .0/ < 1, then, for i  1, ."/

pi

 q ."/ 

..0/ /i .0/ as " ! 0; e iŠ

(6.2.41)

while pNi."/ D

1 X .."/ /j j Di

jŠ

."/

e  q ."/  q ."/  Mi.0/ as " ! 0;

where ."/ Mi

D

1 X .."/ /j j Di

jŠ

."/

e  :

(6.2.42)

6.2

M/G queueing systems with quick service

387

Relation (6.2.42) implies that condition O12 does not hold, since all the probabilities i  1, have the same asymptotic order. Therefore, the following asymptotic relations take place for i D 1; : : : ; N  1: pNi."/ ,

."/ iC1 fi0

 pNi."/  q ."/  Mi.0/ as " ! 0:

(6.2.43)

If .0/ D 1, then, for i  1, ."/

pi

."/

 .."/ /i e  q ."/ 

i as " ! 0; iŠ

(6.2.44)

while ."/

pNi

D

1 X .."/ /j

."/

j Di

."/

."/

e  q ."/  .."/ /i e  q ."/ MN i

jŠ

where ."/ MN i D

1 X .."/ /j i

jŠ

j Di

as " ! 0;

(6.2.45)

! 1 as " ! 0:

Relations (6.2.44) and (6.2.45) imply that condition O12 does not hold, because ."/ ."/ ."/

pNi for i  1, since pi =pNi ! 0 as " ! 0 for i  1. Moreover, in this case, for i  1,

."/ pi

."/ as " ! 0: pNi."/  pNiC1 ."/

Indeed, in this case the identities pNi

."/

."/

pi =pNi

."/

."/

C pNiC1 =pNi

."/

."/

 pNiC1 =pNi

."/

D pi

(6.2.46) ."/

C pNiC1 , i  1, imply that 1 D

as " ! 0.

."/ pNi ,

Thus all the probabilities i  1, have the same asymptotic order, moreover, they are asymptotically equivalent. Therefore, the following asymptotic relations take place for i D 1; : : : ; N  1: ."/ iC1 fi0

."/

."/

."/

 pN1  .."/ /e  q ."/ MN 1

as " ! 0:

(6.2.47)

6.2.5 Conditions for mixed ergodic and limit/large deviation theorems for imbedded semi-Markov processes Let us assume that condition T5 and either condition E05 or E005 hold. As was pointed above, these conditions imply that condition T1 holds for the imbedded semi-Markov process ."/ .t /, t  0.

388

6

Quasi-stationary phenomena in stochastic systems

Conditions T5 and E05 correspond to the pseudo-stationary case. In this case, the .0/ set of recurrent-without-absorption states is X1 D f2; : : : ; N g and the set of non.0/ recurrent-without-absorption states is X2 D f1g. Conditions T5 and E005 correspond to the quasi-stationary case. Here, the set of recurrent-without-absorption states is X1.0/ D f1; : : : ; N g and, for the set of non.0/ recurrent-without-absorption states, we have X2 D ¿. ."/

Condition I4 holds, since the distribution function FN .t / D 1  e t of sojourn time in state N is exponential with parameter  for every "  0. Therefore, ."/ the distribution 0 GN N .t / is not concentrated in zero. Moreover, it is obvious that ."/ ."/ ."/ 0 GN N .0/ D FN .0/ 0 GN 1N .0/ D 0. ."/

Conditions N1 and N2 also hold, since the distribution function FN .t / D 1  e t of sojourn time in state N is exponential with parameter  for every "  0. ."/

The power moments pij Œn, n D 0; 1; : : :, take the following form: ."/ pij Œn

Z D

1

0

."/

s n Qij .ds/

8 ."/ ˆ pij ˆ C1 Œn ˆ ˆ < pN ."/ Œn ij D ˆ nŠ=n ˆ ˆ ˆ : 0

if 1  i  N  1, 2  j  i C 1, if 1  i  N  1, j D 0, if i D N , j D N  1, otherwise,

(6.2.48)

where ."/

Z

1

.s/k s ."/ e G .ds/ kŠ 0 .k C n/Š ."/ p ./ < 1; n D 0; 1; : : : ; D n kŠ kCn

pk Œn D

sn

(6.2.49)

and ."/ pNkC1 Œn

Z

1

kC1 s k s .k C n/Š ."/ .1  G ."/ .s// ds D ./ e pN kŠ n kŠ kCnC1 0   kCn X .k C n/Š ."/ D p ./ < 1; n D 0; 1; : : : : (6.2.50) 1  r n kŠ D

sn

r D0

.k/

These formulas imply that moment conditions M11 holds and condition M13 also holds for every k  1.

6.2

M/G queueing systems with quick service

389

."/

The mixed power-exponential moments pij Œ; n, n D 0; 1; : : :, take the following form for  < : Z 1 ."/ ."/ s n e s Qij .ds/ pij Œ; n D 0 8 ."/ ˆ if 1  i  N  1, 2  j  i C 1, ˆ ˆ pij C1 Œ; n ˆ < pN ."/ Œ; n if 1  i  N  1, j D 0, ij D (6.2.51) nC1 ˆ if i D N , j D N  1, ˆ nŠ=.  / ˆ ˆ : 0 otherwise, where pr."/ Œ; n

Z

1

.s/r s ."/ e G .ds/ rŠ 0 r .r C n/Š ."/ D .  / < 1; p .  /r Cn rŠ rCn

D

s n e s

r; n D 0; 1; : : : ;

(6.2.52)

and ."/ pNr C1 Œ; n

Z D

1

0 r C1 .r 

s n e s

r C1 s r s .1  G ."/ .s// ds e rŠ

C n/Š ."/ pN .  / .  /r CnC1 rŠ rCnC1   rX Cn r C1 .r C n/Š ."/ D 1 pl .  / < 1; .  /r CnC1 rŠ D

r; n D 0; 1; : : : : (6.2.53)

lD0

As was mentioned above, conditions T5 and E05 correspond to the pseudo-stationary case. Formulas (6.2.51), (6.2.52), and (6.2.53) imply that condition C12 holds for any 0 < ı < . In this case, as follows from Lemma 4.5.8, condition C13 also holds. Also, as was mentioned above, conditions T5 and E005 correspond to the quasistationary case. Let us show that conditions G1 and G2 hold with the parameter  D . .0/ In the quasi-stationary case, the set of recurrent-without-absorption states is X1 D .0/ f2; : : : ; N g, and the set of non-recurrent-without-absorption states is X2 D f1g. .0/ .0/ In this case, as follows from formula (6.2.52), (a) pij Œ; 0 D pij C1 Œ; 0 < 1, .0/

 <  for 1  i  N  1, 2  j  i C 1; (b) pN N 1 Œ; 0 D =.  / < 1, .0/ .0/ .0/  < ; (c) pi0 Œ; 0 D pNi Œ; 0 < 1 for 1  i  N  1; and (d) pN N 1 Œ; 0 D =.  / ! 1 as  < ,  ! . Relations (a)–(d) imply that condition G1 holds with the parameter  D .

390

6

Quasi-stationary phenomena in stochastic systems ."/

.0/

Also, relations (6.2.51), (6.2.52), and (6.2.53) imply that (e) pij Œ; 0 ! pij Œ; 0 < 1 as " ! 0 for  <  and i ¤ 0, j 2 X. Relation (e) obviously implies that .0/ ./ D condition G2 (a) holds with the parameter  D . Since, in this case, 0 '12 .0/  p12 Œ; 0 < 1, condition G2 (b) also holds with the parameter  D . As was shown in Lemma 4.5.10, condition C13 is implied by conditions G1 and G2 . In this case, the characteristic root 0 < .0/ <  D . Therefore, by Lemma 4.5.10, (e) conditions C12 and C13 hold for the state i D N and some 0 < ˇN D ˇ < ı <  D  for the imbedded semi-Markov processes ."/ .t /, t  0. Also, by Lemma 4.5.10, (f) conditions C14 –C16 hold for the state i D N and some 0 < ˇN D ˇ < ı <  D  and "07 ; "08 ; "015 > 0 (the parameters used in these conditions) for the imbedded semi-Markov processes ."/ .t /, t  0.

6.2.6 Conditions for mixed ergodic and limit/large deviation theorems for the regenerative processes  ."/ .t/ As was pointed out in Subsection 6.2.2, the subsequent moments of jumps from the state N  1 to the state N are regenerative moments for both processes ."/ .t / and ."/ .t /. ."/ ."/ This implies that the distribution functions 0 GN N .t / and GN N .t /, for the imbedded semi-Markov process ."/ .t /, coincide, respectively, with the distribution functions F ."/ .t / and FO ."/ .t / for the regenerative process ."/ .t /. Also, if the initial state is ."/ .t / D i ¤ 0; N , then the distribution functions ."/ ."/ ."/ 0 GiN .t / and GiN .t / for the imbedded semi-Markov process  .t / coincide, respectively, with the distribution functions FQ ."/ .t / and FOQ ."/ .t / for the regenerative process ."/ .t /. In Chapter 4 it was shown in what way the conditions used in Chapter 3 for regenerative processes are employed as the corresponding conditions used in Chapter 4 for semi-Markov processes when considering these processes as regenerative processes with return times in some recurrent state that is regarded as regeneration times. By applying these results to the semi-Markov processes ."/ .t / we get that T5 and E05 imply that conditions D13 , M8 , and D17 –D20 hold as well as conditions C3 and C8 . Also, conditions T5 imply that conditions D21 , C6 hold as well as conditions D23 , C10 , and C11. The only conditions that require special considerations are F7 –F9 . This is because ."/ the forcing functions PN f ."/ .t / D j; 1 ^ ."/ > t g, t  0, and PN f."/ .t / D j ; ."/

1 ^ ."/ > t g, t  0, for the corresponding renewal equations, which can be written for distributions of the regenerative process ."/ .t / and the imbedded semi-Markov process ."/ .t /, are not the same. For i; j D 1; : : : ; N , denote ."/

."/

qij .t / D Pi f ."/ .t / D j; 1 ^ ."/ > t g;

t  0:

6.2

M/G queueing systems with quick service

391

In order to check condition F7 , we shall prove that (a) condition T5 implies the local ."/ .0/ .t / to the function qij .t / as " ! 0 almost uniform convergence of the functions qij everywhere with respect to the Lebesgue measure on Œ0; 1/ for every i; j D 1; : : : ; N . ."/ ."/ Let us use the random variables ˛0 D 0 and ˛n , n D 1; : : :, introduced in ."/ ."/ ."/ Subsection 6.2.2. Also define the random variables #n D ˛1 C    C ˛n , n D ."/ 0; 1; : : :, and also let 0N D min.n  1 W ."/ .#n."/ / 2 f0; N g/. Using these random variables we can write the following formula for the probabil."/ ities qij .t / for i D 1; : : : ; N , j D 1; : : : ; N  1: ."/ qij .t / D

1 N 1 X X

."/ Pi f0N > n; #n."/  t; ."/ .#n."/ / D k;

kDj nD0 ."/ > t;  ."/ .t /   ."/ .#n."/ / D k  j g; #nC1

t  0: (6.2.54)

Also, for i D 1; : : : ; N , j D N , ."/

qiN .t / D ı.i; N /e t ;

t  0:

(6.2.55)

."/ Formulas (6.2.54) and (6.2.55) show that the probability qij .t /, if regarded as a

."/ of discontinuity points for function of t  0, has not more than a countable set Rij every i; j D 1; : : : ; N . According to the remarks made in Subsections 3.2.3 and 3.3.3, this implies that conditions F8 and F9 hold, since conditions M8 and C6 hold. Let us prove that (b) condition T5 implies the local uniform convergence of the ."/ .0/ .0/ functions qij .t / to the function qij .t / as " ! 0, t 2 Sij , for every i; j D 1; : : : ; N , .0/ .0/ where SNij D Œ0; 1/ n Sij , i; j D 1; : : : ; N , are some at most countable subsets of Œ0; 1/. Since the sets SNij.0/ , i; j D 1; : : : ; N , are at most countable, statement (b) implies (a). Formula (6.2.55) obviously implies that statement (b) holds for i D 1; : : : ; N , j D N. The following obvious formula also permits to reduce the proof of statement (b) for i D N , j D 1; : : : ; N  1 to the case where i D N  1, j D 1; : : : ; N  1, ."/ qNj .t /

Z D

t 0

."/

qN 1j .t  s/e s ds;

t  0:

(6.2.56)

."/ .0/ Statement (b) for the functions qN 1j .t /, t  0, means that there exists a set SN 1j such that its complement, SN .0/ , is a countable set and, for any t" ! t0 as " ! 0, .0/

N 1j ."/

.0/

where t0 2 SN 1j , we have qN 1j .t" / ! qN 1j .t0 / as " ! 0.

392

6

Quasi-stationary phenomena in stochastic systems

Let us choose T > t0 . There exists "0 > 0 such that t"  T for "  "0 . Let us also ."/ define functions qN 1j .s/ D 0 for s < 0. Now, by applying formula (6.2.56), the Lebesgue theorem, and statement (b) to the ."/ functions qN 1j .t /, we get ."/ qNj .t" /

Z D

T 0

Z !

T 0

."/ s qN ds 1j .t"  s/e .0/

.0/

qN 1j .t0  s/e s ds D qNj .t0 / as " ! 0:

(6.2.57)

."/

.0/

Thus statement (b) also holds for the functions qNj .t /, t  0. The set SNj D

.0/

SN 1j can be taken as a corresponding set of local uniform convergence. Therefore, it only remains to consider the cases i; j D 1; : : : ; N  1. Let us first consider a more simple case where conditions T5 and E05 hold. In this case, the following simple estimate can be used for i; j D 1; : : : ; N  1: ."/

."/

."/

qij .t /  Pi f."/ ^ 1 > t g D 1  GiN .t /;

t  0:

(6.2.58)

As was pointed out in Subsection 6.2.2, conditions T5 and E05 imply that all limit .0/ distributions of sojourn times, Fj .t /, j D 1; : : : ; N  1, are concentrated in 0. ."/

Therefore, the distributions GiN .t /, i D 1; : : : ; N  1, are also concentrated in 0, i.e., ."/ 1  GiN .t / D 0, t  0, for i D 1; : : : ; N  1. On the other hand, by Lemma 4.2.3, ."/ .0/ ./ ) GiN ./ as " ! 0 for i D 1; : : : ; N  1 that, in this case, is equivalent to GiN ."/ .0/ .t / ! 1  GiN .t / D 0 as " ! 0 for t > 0 and i D 1; : : : ; N  1. the relation 1  GiN ."/ Since the functions 1  GiN .t / are non-increasing, the last relation implies that (c) the ."/

.0/

functions 1  GiN .t / locally uniformly converge to the function 1  GiN .t / D 0 as " ! 0 for any t > 0 and i D 1; : : : ; N  1. ."/ .t / locally Due to estimate (6.2.58), statement (c) implies that (d) the functions qij .0/

uniformly converge to the function qij .t / D 0 as " ! 0 for any t > 0 and i; j D .0/

1; : : : ; N  1. Thus statement (b) holds true with the sets Sij D .0; 1/, i; j D 1; : : : ; N  1. Let us now consider a more complex case where conditions T5 and E005 hold. Introduce Markov renewal functions for i; j D 1; : : : ; N  1 by ."/

Uij .t / D

1 X

."/

Uij .n; t /;

t  0;

nD0

where

."/

."/

Uij .n; t / D Pi f0N > n; #n."/  t; ."/ .#n."/ / D j g;

t  0:

6.2

393

M/G queueing systems with quick service

By the definition, for i; j D 1; : : : ; N  1, ."/

."/

."/

Uij .0; t / D ı.i; j /; Uij .1; t / D Qij .t /;

t  0:

(6.2.59)

."/

Also, the functions Uij .n; t / satisfy the following recurrence convolution relations for i; j D 1; : : : ; N  1 and n D 1; 2; : : :: N 1 Z t X ."/ ."/ ."/ Ukj .n  1; t  s/Qik .ds/; t  0: (6.2.60) Uij .n; t / D kD1

0

Relations (6.2.11), (6.2.59), and (6.2.60) imply that the following weak conver."/ gence relations hold for the functions Uij .n; t / (which are, in fact, improper distribution functions in the argument t  0), for n D 0; 1; : : :, i; j D 1; : : : ; N  1: ."/

.0/

Uij .n; / ) Uij .n; / as " ! 0: ."/

(6.2.61)

."/

Let us denote by  ."/ .t / D max.n W &1 C    C &n  t /, t  0, the renewal ."/ process with inter-renewal times &n , n D 1; 2; : : :, which are subsequent service times. Note that (m) in the case where the initial state is ."/ .0/ D i ¤ 0; N , the random ."/ ."/ ."/ ."/ event satisfies f0N > n; #n  t g f&1 C    C &n  t g D f ."/ .t /  ng for t  0 and n D 0; 1; : : : . The following asymptotic estimate follows from conditions T5 and E005 , the corresponding estimate (1.2.55) for moments of the renewal process, and relations (m) for t  0 and n D 0; 1; : : :: lim lim

n!1 "!0

1 X

."/

Uij .n; t /  lim lim

n!1 "!0

kDn

1 X

Pf ."/ .t /  kg

kDn

 lim E ."/ .t /2 lim

n!1

"!0

1 X

1=k 2 D 0:

(6.2.62)

kDn ."/

Relations (6.2.61) and (6.2.62) imply that the renewal functions Uij .t / generate, on the Borel -algebra of interval Œ0; 1/, measures that take finite values on finite intervals, Uij."/ ..u; v/ D Uij."/ .v/  Uij."/ .u/ < 1; 0  u  v < 1, and these measures weakly converge for i; j D 1; : : : ; N  1, ."/

.0/

Uij ./ ) Uij ./ as " ! 0:

(6.2.63)

Now, using the independence of the service processes and the input flow, and the ."/ Markov property of the process ."/ .t / at the moments #n , we can re-write formula (6.2.58) in the following form for i; j D 1; : : : ; N  1: N 1 Z t X ..t  s//kj .ts/ ."/ ."/ e .1  G ."/ .t  s//Uik .ds/; t  0: (6.2.64) qij .t / D .k  j /Š 0 kDj

394

6

Quasi-stationary phenomena in stochastic systems

.0/

.0/

Let also Si be the set of all points t0 2 Œ0; 1/ such that (1) the measures Uik .A/, k D 1; : : : ; N  1, have no atom at the point t0 ; (2) every point t0  s, where s is some .0/ point in which at least one of the measures Uik .A/, k D 1; : : : ; N 1, has an atom, is a .0/ point of continuity of the function 1G .u/. Obviously, the set SNi.0/ D Œ0; 1/nSi.0/ is at most countable. ."/ In order to prove statement (b) for the functions qij .t /, we should prove that ."/

.0/

.0/

qij .t" / ! qij .t0 / as " ! 0 for any t" ! t0 as " ! 0, where t0 2 Si . Let us choose T > t0 such that the point T is a point of continuity of all renewal .0/ functions Uik .t /, k D 1; : : : ; N  1. There exists "0 > 0 such that t"  T for "  "0 .

Conditions T5 and E005 imply that (e) 1  G ."/ .t"  s/ ! 1  Gij.0/ .t0  s/ as " ! 0 for every s such that the point t0  s is a continuity point of the limit function. Since the functions 1  G ."/ .t"  s/ are non-increasing in the argument s, statement (e) implies that (f) the functions 1  G ."/ .t"  s/ converge locally uniformly to the function 1  G .0/ .t0  s/ as " ! 0 at every point of continuity of the limit function. .0/ The assumptions that t0 2 Si and T is a point of continuity of all renewal func.0/ tions Uik .t /, k D 1; : : : ; N  1, the relation of weak convergence (6.2.63), and state."/

ment (f) permit to apply Lemma 1.2.2 to the measures Uik .A \ Œ0; T / and the funcs//kj

" tions ..t.kj e .t" s/ .1G ."/ .t" s//.s  t" / for every i; k; j D 1; : : : ; N 1. /Š This yields the following relation:

."/ .t" / D qij

N 1 Z T X kDj

0

..t"  s//kj .k  j /Š ."/

 e .t" s/ .1  G ."/ .t"  s//.s  t" /Uik .ds/ N 1 Z T X ..t0  s//kj .0/ ! qij .t0 / D .k  j /Š 0 kDj

.0/

 e .t0 s/ .1  G .0/ .t0  s//.s  t0 /Uik .ds/ as " ! 0: (6.2.65) ."/

.0/

.0/

Thus statement (b) holds for the function qij .t / with the set Sij D Si i; j D 1; : : : ; N  1.

for every

6.2.7 Quasi-stationary distributions for the regenerative processes  ."/ .t/ We will use the moment generation functions for i D 1; : : : ; N , Z 1 Z 1 ."/ ."/ ."/ ."/ t  ./ D e G .dt /;  ./ D e t GiN .dt /; 0 iN 0 iN iN 0

0

As was mentioned above, T5 implies that conditions C12 –C16 hold for the imbedded semi-Markov processes ."/ .t / with some ˇ < ı <  and, therefore, by Lemmas 4.5.6

6.2

395

M/G queueing systems with quick service ."/

and 4.5.7, there exists "0 > 0 such that, for "  "0 , (a) 0 N N .ˇ/ 2 .1; 1/; (b) ."/ ."/ 0 iN .ˇ/ < 1, i D 1; : : : ; N ; (c) iN .ˇ/ < 1, i D 1; : : : ; N . Let us also introduce moment generating functions by ."/ ./ D !iN

Z

1 0

."/ e t .1  GiN .t // dt;

."/ !iNj ./ D

Z

1 0

."/ e t qij .t / dt:

Obviously, ."/ ./ D !iN

N X

."/ !iNj ./:

(6.2.66)

j D1 ."/

."/

Note that (d) 1 C !iN ./ D iN ./ for i D 1; : : : ; N . This relation implies that ."/ ."/ .ˇ/ < 1, i D 1; : : : ; N ; (f) !iNj .ˇ/ < 1, i; j D 1; : : : ; N . (e) !iN Formula (3.3.11) found for the quasi-stationary distribution of regenerative process given in Subsection 3.3.3, if applied to the regenerative process ."/ .t /, gives the following formula for its quasi-stationary distribution: ."/

j."/ .."/ /

D

!N Nj .."/ / ."/

!N N .."/ /

;

j D 1; : : : ; N;

(6.2.67)

where ."/ < ˇ is the unique root of the characteristic equation ."/ 0 N N ./

D 1:

(6.2.68)

Formula (6.2.67) for the quasi-stationary distribution of the regenerative process ."/ .t / is similar to the corresponding variant of formula (5.5.36) for the quasi-stationary distribution specified for the imbedded semi-Markov process ."/ .t /. In these ."/ formulas, !N Nj .."/ /, j ¤ 0, are, respectively, the mixed power-exponential moment of the generating functions for the times spend by the regenerative process ."/ .t / or the imbedded semi-Markov process ."/ .t / in a given state before hitting the state N or being absorbed. Let us introduce the following mixed power-exponential moment generating functions for times spend by the process ."/ .t / in a given state before hitting the state N or being absorbed for i; j D 1; : : : ; N , n D 0; 1; : : :, and  < : Z

."/

'ij Œ; n D Ei Z D

0

."/

˛1

0 1

s n e s . ."/ .s/ D j / ds ."/

s n e s Pi f ."/ .s/ D j; ˛1 > sg ds:

396

6

Quasi-stationary phenomena in stochastic systems

The following formula takes place for i; j D 1; : : : ; N and n D 0; 1; : : :, and n D 0; 1; : : :: ."/

'ij Œ; n D 8 ."/ i j .ij Cn/Š ˆ Nij ˆ i j C1 .ij /Š p Cn .  / ./ ˆ ˆ < 0 ˆ 0 ˆ ˆ ˆ : nŠ ; ./nC1

if 1  i  N  1, 1  j  i , if 1  i  N  1, i C 1  j  N , (6.2.69) if i D N , 1  j  N  1, if i D N , j D N .

Indeed, formula (6.2.70) is obvious in the cases i D N and 1  i  N  1, i C 1  j  N , while for the case 1  i  N  1, 1  j  i , Z 1 ."/ ."/ s n e s Pi f ."/ .s/ D j; ˛1 > sg ds 'ij Œ; n D Z D

0

1 0

sn

.s/ij ./s e .1  G ."/ .s// ds .i  j /Š

ij .i  j C n/Š ."/ pN .  /: .  /ij CnC1 .i  j /Š ij Cn

D

(6.2.70)

The following representation can be obtained for j ¤ 0: Z

."/

."/ ^1 0

e s . ."/ .s/ D j / ds Z D

."/

˛1

e s . ."/ .s/ D j / ds C

0

X

. ."/ .˛1."/ / D i /

i¤0;N ."/

 e ˛1

Z

."/ ."/ ."/ ^˛1 1

0

."/

e s . ."/ .˛1 C s/ D j / ds:

(6.2.71)

Taking expectations on the left- and the right-hand sides of (6.2.71) we get the fol."/ lowing system of linear equations for the moment generating functions !iNj ./; i D 1; : : : ; N , for every   ˇ; "  "0 , and j D 1; : : : ; N : ."/

."/

!iNj ./ D 'ij Œ; 0 C

X

."/

."/

pir Œ; 0!rNj ./;

i ¤ 0:

(6.2.72)

r ¤0;N

This system of linear equations should be compared with the system of linear equations for mixed power-exponential moment generating functions for the times spend by the imbedded semi-Markov process ."/ .t / in a given state before hitting the state N or an absorption.

6.2

M/G queueing systems with quick service

397

Both systems have the same coefficient matrix N P."/ ./ and differ only in the free terms. In the case of the imbedded semi-Markov process, the free terms 'ij."/ Œ; 0, i D P ."/ ."/ 1; : : : ; N , should be replaced with the free terms 'Qij Œ; 0 D ı.i; j / irD1 'ir Œ; 0, i D 1; : : : ; N . This is because the imbedded semi-Markov process ."/ .t / for the process ."/ .t / is defined in such a way that Pi f."/ .s/ D j; ˛1."/ > sg D ı.i; j /  Pi ."/ ."/ r D1 Pi f .s/ D r; ˛1 > sg for every s  0 and i; j D 1; : : : ; N . Both systems have unique solutions for every   ˇ and "  "0 . A detailed discussion concerned the latter system can be found in Section 5.5.7, while the corresponding threshold value "0 is determined in Subsection 4.5.4. In the quasi-stationary case, where conditions T5 and E005 hold, the limit quasi.0/ stationary probabilities j ..0/ /, j D 1; : : : ; N , are positive and can be found using formulas (6.2.66) and (6.2.67). The corresponding moment generating function .0/ .0/ !N Nj . /, j D 1; : : : ; N , can be found by solving the system of linear equations (6.2.72), which should be solved for the value  D .0/ > 0 which is a solution of equation (6.2.68). Note also that, in this case, the transition quantities are positive, .0/ .0/ 0 iN . / > 0, j D 1; : : : ; N  1. In a simpler pseudo-stationary case, where conditions T5 and E05 hold, the limit .0/ quasi-stationary probabilities satisfy j ..0/ / D ı.j; N /, j D 1; : : : ; N . In this case, .0/ D 0 and, as follows from relations (6.2.55) and (6.2.56) and statement .0/ (l) formulated in Subsection 6.2.6, we have qNj .t / D ı.j; N /e t , t  0, for j D 1; : : : ; N . Note also that, in this case, the transition stopping probabilities satisfy .0/ 0 GiN .1/ D 1, j D 1; : : : ; N  1. So, all conditions introduced in Chapter 3 are verified. All the results given in Chapter 3 are applicable to the regenerative process ."/ .t /, t  0, which describes functioning of M/G queueing systems with quick service and a bounded queue buffer.

6.2.8 Mixed ergodic and limit/large deviation theorems Conditions T5 and E05 correspond to a pseudo-stationary model with the absorption probabilities asymptotically vanishing to zero. Under these conditions there are analogues of mixed ergodic and limit Theorems 3.2.1 and 3.2.5 and mixed ergodic and large deviation Theorems 3.3.1 and 3.3.8 covering the case of the pseudo-stationary asymptotics. T5 and E005 correspond to a quasi-stationary model with absorption probabilities asymptotically separated from zero. Under these conditions, analogues of mixed ergodic and large deviation Theorems 3.3.2–3.3.4 and 3.3.7–3.3.12 covering the case of quasi-stationary asymptotics can be formulated, together with Theorem 3.3.5, 3.3.6, and 3.3.13, which give conditions for convergence of the corresponding quasi-stationary distributions. It is useful to note that the theorems listed above cover the pseudo-stationary case, where conditions T5 and E005 hold as well.

398

6

Quasi-stationary phenomena in stochastic systems

The limit parameters, including the stationary and quasi-stationary distributions, can be calculated by using the corresponding algorithms and formulas given in Chapter 4. They should be specified for the semi-Markov processes ."/ .t /, t  0, introduced in the current section.

6.3 Exponential asymptotics for M/G queueing systems with quick service In this section, we continue to study M/G queueing systems with quick service and nonlinearly perturbed parameters.

6.3.1 Perturbation conditions in mixed ergodic and large deviation theorems Let us formulate perturbation conditions that replace the corresponding perturbation conditions introduced in Chapters 3 and 5 in mixed ergodic and limit/large deviation theorems. Let us introduce the following perturbation condition for  < : .;k/

P32

."/

.0/

: pn .  / D pn .  / C en Œ; 1" C    C en Œ; k"k C o."k / for n D 0; : : : ; N C k, where jen Œ; lj < 1, n D 0; : : : ; N C k, l D 1; : : : ; k.

It is also convenient to define en Œ; 0 D pn.0/ .  / for n D 0; : : : ; N C k. .;k/ Note that condition P32 implies that the following asymptotic expansions take place for n D 0; : : : ; N C k: ."/

pNnC1 .  / D 1 

n X

."/ pm .  /

mD0

D

.0/ pNnC1 .

 / C eNnC1 Œ; 1" C    C eNnC1 Œ; k"k C o."k /; (6.3.1)

.0/

where eNn Œ; 0 D pNn .  / for n D 0; : : : ; N C k, and, for n D 0; : : : ; N C k, l D 0; : : : ; k, ( P 1  nmD0 em Œ; l if l D 0; (6.3.2) eNnC1 Œ; l D P if l D 1; : : : ; k.  nmD0 em Œ; l These conditions should be used for the value  D 0 in the pseudo-stationary case, i.e., if conditions T5 and E05 hold, and for the value  D .0/ > 0, which is a solution of equation (6.2.68), in the quasi-stationary case, i.e., if conditions T5 and E005 hold. As an example, let us consider a model for which condition T6 holds. In this case, it .;k/ is natural to replace the perturbation condition P32 with the following perturbation

399

6.3 Exponential asymptotics for M/G queueing systems with quick service

condition imposed on the parameter ."/ : .k/

P33 : ."/ D .0/ C 1 " C    C k "k C o."k /, where jr j < 1, r D 1; : : : ; k. It is also convenient to define 0 D .0/ . In this model, for n D 0; 1; : : :, Z 1 ..  /s/n ./s pn."/ .  / D G.ds=."/ / e nŠ 0 Z 1 ..  /s/n ./."/ s ."/ n D . / G.ds/: e nŠ 0

(6.3.3)

Let us consider the pseudo-stationary case, where condition E06 holds, i.e., .0/ D 0. Let us first consider the case where ."/ D O."/, i.e., the following condition also holds: O17 : 1 > 0. .k/

.NCk/

imply that Let us show that, in this case, conditions P33 , E06 , O17 , and M14 .;k/ condition P32 holds for every  <  with the coefficients in the corresponding expansions, 8 ˆ if 0  l  k, k < n  N C k, 0. In this case, formulas (6.3.9) and (6.3.10) yield that en Œ; l D 0 for l < nh. Now, let us consider the quasi-stationary case where condition E006 holds, i.e., .0/ > 0. .k/ .;k/ Let us show that, in this case, conditions P33 and E006 imply that condition P32 holds for every  <  with the coefficients in the corresponding expansions given below by formulas (6.3.18) and (6.3.19). ."/ Denote 1 D ."/  .0/ . Since .0/ > 0 and ."/ ! .0/ as " ! 0, there exists .0/ .0/ ."/ "0 < 0 such that j1 j  2 and, therefore, ."/  2 for "  "0 .

402

6

Quasi-stationary phenomena in stochastic systems

In this case, formula (6.3.3) implies that Z 1 ..  /s/n ./.0/ s ."/ ."/ n pn .  / D . / e nŠ 0  X k ."/ ..  /1 s/r G.ds/  rŠ r D0 Z 1 ..  /s/nCkC1 ./.0/ s ."/ C .1 /kC1 .."/ /n e nŠ 0   X 1 ."/ ..  /1 s/rk1 G.ds/: (6.3.14)  rŠ r DkC1

Note that ˇ 1 ˇ 1 ."/ r k1 ˇ X ˇ X ..  /."/ ..  /j1 js/rk1 1 s/ ˇ ˇ ˇ ˇ rŠ rŠ r DkC1

rDkC1

."/

 e ./j1 and, for any m D 0; 1; : : : and ı > 0, Z Mm .ı/ D

1

0

js

(6.3.15)

;

s m e ıs G.ds/ < 1:

(6.3.16)

Thus, by continuing the estimate in (6.3.6) and using estimate (6.3.15), we can conclude that there exists 1   ."/  1 such that, for "  "0 , pn."/ .  / D .."/ /n

k X

."/

.1 /r

r D0

.  /nCr .1/r MnCr ..  /.0/ / nŠrŠ

C .."/ /kC1  ."/

 .  /nCkC1 .0/  MnCkC1 .  / nŠ 2

D en Œ; 0 C en Œ; 1" C    C en Œ; k"k C o."k /;

(6.3.17)

where the coefficients en Œ; l, l D 0; : : : ; k, are given by the formulas en Œ; l D

n X k X

Cnm ..0/ /nm .1/r

mD0 r D0

 MnCr ..  /.0/ / 

.  /nCr .n C r/Š nŠrŠ X

k Y nq ; nq Š

(6.3.18)

n1 ;:::;nk 2DnCr;l qD1

where DnCr;l is the set of all nonnegative integer solutions of the system n1 C    C nk D n C r;

n1 C    C k nk D l:

(6.3.19)

6.3 Exponential asymptotics for M/G queueing systems with quick service

403

6.3.2 Asymptotic expansions for mixed power-exponential moments for hitting times for the imbedded semi-Markov processes Consider the mixed power-exponential generation functions for  2 R1 and i D 1; : : : ; N , Z 1 ."/ ."/  Œ; n D t n e t 0 GiN .dt /: 0 iN 0

."/

As was shown in Subsection 5.3.5, relation 0 iN .ˇ/ < 1, i D 1; : : : ; N , implies ."/ that (a) 0 iN Œ; n < 1, n D 0; 1; : : :, i D 1; : : : ; N , for  < ˇ. To construct asymptotic expansions for the characteristic root ."/ of equation (6.2.68) we should, according to the results given in Chapters 3 and 5, use the asymp."/ totic expansions for the mixed power-exponential generation functions 0 iN Œ; n, i ¤ 0, n D 0; : : : ; k. These expansions are based on the perturbation condition .;k/ P25 , which in its turn is based on the asymptotic expansions for the mixed power."/ exponential generation functions pij Œ; n, i; j ¤ 0, n D 0; : : : ; k. The corresponding algorithm is described in Section 5.2. .;k/ and formula (6.2.52) imply that the followNow, the perturbation condition P32 ing asymptotic expansions take place for r D 1; : : : ; N , n D 0; : : : ; k: pr."/ Œ; n D pr.0/ Œ; n C er Œ; 1; n" C    C er Œ; k; n"k C o."k /;

(6.3.20)

.0/

where pr Œ; n D er Œ; 0; n for n D 0; : : : ; k and, for r D 1; : : : ; N , n; l D 0; : : : ; k, er Œ; l; n D

r .r C n/Š erCn Œ; l: .  /r Cn rŠ

(6.3.21)

Formula (6.2.51) and the asymptotic relations (6.3.20) imply that the following asymptotic expansions take place for i; j ¤ 0, n D 0; : : : ; k: ."/

.0/

pij Œ; n D pij Œ; n C eij Œ; 1; n" C    C eij Œ; k; n"k C o."k /;

(6.3.22)

.0/

where pij Œ; n D eij Œ; 0; n, for n D 0; : : : ; k and, for i; j ¤ 0, n; l D 0; : : : ; k, 8 ˆ < eij C1 Œ; l; n nŠ eij Œ; l; n D ı.l; 0/ ./ nC1 ˆ : 0

if 1  i  N  1, 2  j  i C 1, if i D N , j D N  1,

(6.3.23)

otherwise. .;k/

Therefore, the perturbation condition P32 implies that the perturbation condition .;k/ P25 holds, which, in this case, is represented by the asymptotic relations (6.3.22). As was mentioned above, this condition permits to construct the asymptotic expansions

404

6

Quasi-stationary phenomena in stochastic systems ."/

for the power moments 0 iN Œ0; n, i ¤ 0, n D 0; : : : ; k, which constitute the perturN

bation conditions P9.k/ and P.10k/ , or to construct the asymptotic expansions for mixed ."/ .0/ power-exponential moments 0 iN Œ ; n, i ¤ 0, n D 0; : : : ; k, which constitute the .kN /

.k/

perturbation conditions P11 and P12 .

6.3.3 Asymptotic expansions for mixed power-exponential moments for the time spent by the process  ."/ .t/ in a given state before hitting a given state or an absorption Let us return to the system of linear equations (6.2.72). As was pointed out in Subsec."/ tion 6.2.7, (a) 'ij."/ Œ; 0 < 1 for i; j ¤ 0, for  <  and "  0; and (b) !iNj ./ < 1, i; j ¤ 0 for  < ˇ and "  "0 . ."/ Relations (a) and (b) imply that (c) 'ij Œ; n < 1 for i; j ¤ 0, n D 0; 1; : : :, and ."/

"  0; (d) !iNj ./ < 1, i; j ¤ 0, n D 0; 1; : : :, for  < ˇ and "  "0 , and that (e) we can differentiate (with respect to the variable ) equations of system (6.2.72) and get the following system of linear equation (which coincides with system (6.2.72) for n D 0) for every  < ˇ, "  "0 , j ¤ 0, and n D 0; 1; : : :: X ."/ ."/ ."/ ."/ pir ./!rNj Œ; n; i ¤ 0; (6.3.24) !iNj Œ; n D #iNj Œ; n C r ¤0;N ."/ ."/ where !iNj Œ; 0 D !iNj ./, i; j ¤ 0 and, for i; j ¤ 0, n D 0; 1; : : :, ."/

."/

#iNj Œ; n D 'ij Œ; n C

n X X

."/

."/

Cnm pir Œ; m!rNj Œ; n  m:

(6.3.25)

r ¤0;N mD1

The systems of linear equation (6.3.24) can be rewritten in a matrix form. Introduce the vectors for  < ˇ; "  "0 , j ¤ 0, and n D 0; 1; : : : by 2 ."/ 2 ."/ 3 3 !1Nj Œ; n #1Nj Œ; n 6 6 7 7 :: :: 6 6 7; 7; ¨."/ ª ."/ : : Nj Œ; n D 4 Nj Œ; n D 4 5 5 ."/ ."/ !N Nj Œ; n #N Nj Œ; n and

2

3 ."/ '1j Œ; n 6 7 ."/ :: 7: ®j Œ; n D 6 : 4 5 ."/ 'Nj Œ; n

Then the system of linear equations (6.3.24) can be rewritten in the following form for every  < ˇ, "  "0 , j ¤ 0, and n D 0; 1; : : :: ."/ ."/ ."/ ¨."/ Nj Œ; n D ª Nj Œ; n C N P Œ; n ¨Nj Œ; n;

(6.3.26)

6.3 Exponential asymptotics for M/G queueing systems with quick service

405

where the matrices N P."/ Œ; n have the following form for n D 0; 1; : : :: 2

p0."/ Œ; n ."/ p1 Œ; n :: :

0 0 :: :

0 ."/ p0 Œ; n :: :

::: ::: :: :

0 0 :: :

0 0 :: :

6 6 6 ."/ 6 P Œ; n D N 6 6 ."/ ."/ ."/ 4 0 pN 2 Œ; n pN 3 Œ; n : : : p1 Œ; n 0 nŠ 0 0 0 : : : ./ 0 nC1

3 7 7 7 7: 7 7 5

(6.3.27)

The systems of linear equations (6.3.26) are systems of -hitting type with the same coefficient matrix N P."/ Œ; 0. Under conditions T5 , these systems have unique solutions for every "  "0 ,  < ˇ, j ¤ 0, and n D 0; 1; : : : . These solutions have the following forms: ."/ 1 ."/ ¨."/ Nj Œ; n D ŒI  N P Œ; 0 ª Nj Œ; n:

(6.3.28)

Systems (6.3.26) have, for a given j ¤ 0, the same coefficient matrix N P."/ ./ but ."/ different inhomogeneous terms ª Nj Œ; n for n D 0; 1; : : : . These systems should be solved recursively. First, system (6.3.26) should be solved for the case n D 0. Second, system (6.3.26) should be solved for the case n D 1. Note that the expressions for the inhomogeneous P ."/ ."/ ."/ Œ; 1 D 'ij."/ Œ; 1 C r ¤0;N pir Œ; 1!rNj Œ; 0, i ¤ 0, given in (6.3.25) terms iNj ."/

include the solutions !iNj Œ; 0, i ¤ 0, of systems (6.3.26) for n D 0. This recursive procedure should be repeated for n D 1; 2; : : : . The expressions for ."/ ."/ the inhomogeneous terms #ij l Œ; n given in (6.3.25) include the solutions !ij l Œ; m, i ¤ 0, of systems (6.3.26) for m D 0; 1; : : : ; n  1. Formula (6.3.28) can be rewritten in the following vector form for every  < ˇ, "  "0 , and n D 0; 1; : : : ; j ¤ 0: ."/

."/

ª Nj Œ; n D ®j Œ; n C

n X

."/

Cnm N P."/ Œ; m¨Nj Œ; n  m:

(6.3.29)

mD1

Using formula (6.3.29) we can re-write formula (6.3.28) in the following form representing the recursive procedure for calculating solutions of the system of linear equations (6.3.26) for  < ˇ, "  "0 , and n D 0; 1; : : :, j ¤ 0, ."/ 1 ."/ ¨."/ Nj Œ; n D ŒI  N P ./ ª Nj Œ; n

D ŒI  N P."/ ./1 ®."/ j Œ; n C

n X mD1

Cnm ŒI  N P."/ ./1 N P."/ Œ; m¨."/ Nj Œ; n  m: (6.3.30)

406

6

Quasi-stationary phenomena in stochastic systems ."/

We are now in a position to give asymptotic expansions for the functions !iNj Œ; n, i; j ¤ 0, n D 0; 1; : : : ; k. .;k/ Formula (6.2.69) and the perturbation condition P32 imply the following asymp."/ totic expansion for the vectors ®j Œ; n, j ¤ 0, n D 0; : : : ; k, i; j ¤ 0, n D 0; : : : ; k: ."/

.0/

'ij Œ; n D 'ij Œ; n C wij Œ; 1; n" C    C wij Œ; k; n"k C o."k /;

(6.3.31)

.0/

where 'ij Œ; n D wij Œ; 0; n for i; j ¤ 0, n D 0; : : : ; k, and, wij Œ; l; n are given for i; j ¤ 0, n; l D 0; : : : ; k by the following formulas: wij Œ; l; n D 8 i j .ij Cn/Š ˆ eN .  / ˆ ./i j C1 .ij /Š ij Cn ˆ ˆ < 0 ˆ 0 ˆ ˆ ˆ : ı.l; 0/ nŠ ./nC1

if 1  i  N  1, 1  j  i , if 1  i  N  1, i C 1  j  N , (6.3.32) if i D N , 1  j  N  1, if i D N; j D N .

Note also that the asymptotic relation (6.3.20) implies that the following matrix asymptotic expansion can be written for the matrices N P."/ Œ; n for n D 0; 1; : : :: NP

."/

Œ; n D

NP

.0/

where N P.0/ Œ; n D NE

."/

Œ; n C N EŒ; 1; n" C    C N EŒ; k; n"k C o."k /; (6.3.33)

N EŒ; 0; n

for n D 0; : : : ; k, and, for l; n D 0; : : : ; k,

Œ; l; n D 2 0 0 e0 Œ; l; n 6 0 e Œ; l; n e Œ; l; n 1 0 6 :: :: 6 :: 6 : : : 6 4 0 eN 2 Œ; l; n eN 3 Œ; ln 0 0 0

::: ::: :: :

0 0 :: :

::: e1 Œ; l; n nŠ : : : ı.l; 0/ ./ nC1

0 0 :: :

3

7 7 7 7: 7 0 5 0

(6.3.34)

Let us introduce the vectors for j ¤ 0, l; n D 0; : : : ; k, 3 3 2 2 c1Nj Œ; l; n w1j Œ; l; n 7 7 6 6 :: :: wj Œ; l; n D 4 cNj Œ; l; n D 4 5; 5: : : cN Nj Œ; l; n wNj Œ; l; n The following asymptotic relations are corollaries of Lemma 5.3.3 applied to the ."/ functions ¨Nj Œ; n for j ¤ 0 and n D 0; 1; : : : ; k. Proofs of these relations are analogous to those given in Lemmas 5.3.5–5.3.6 and 5.5.7. Thus, under conditions T5 .;k/ and P32 , .0/ k k ¨."/ Nj Œ; n D ¨Nj Œ; n C cNj Œ; 1; n" C    C cNj Œ; k; n" C o." /;

(6.3.35)

6.3 Exponential asymptotics for M/G queueing systems with quick service

407

where the vector coefficients cNj Œ; r; n are given by the recurrence formulas .0/ 1 .0/ cNjŒ; 0; 0 D ¨.0/ Nj Œ; 0 D ŒI  N P ./ ™Nj Œ; 0 and, in general, for r D 0; : : : ; k (for a given n) and subsequently for n D 0; : : : ; k, cNj Œ; r; n D ŒI  N P.0/ ./1 .wj Œ; r; n C

n X

Cnm

mD1

C

r X

r X

N EŒ; q; mcNj Œ; r

 q; n  m

qD0

EN Œ; p; 0cNj Œ; r  p; n/:

(6.3.36)

pD1

The asymptotic relation (6.3.35) allows to construct, in an obvious way, asymptotic expansions for mixed power-exponential moments for A X0 D fj ¤ 0g, i ¤ 0, n D 0; 1; : : :, X ."/ ."/ !iNA Œ; n D !iNj Œ; n; j 2A

which has the following form: ."/

.0/

!iNA Œ; n D !iNA Œ; n C ciNA Œ; 1; n" C    C ciNA Œ; k; n"k C o."k /; (6.3.37) .0/

where !iNA Œ; n D ciNA Œ; 0; n, n D 0; : : : ; k, and, for A X0 D fj ¤ 0g, i ¤ 0, l; n D 0; : : : ; k, X ciNA Œ; l; n D ciNj Œ; l; n: j 2A .;k/

Finally, we conclude that conditions T5 , P32 , and E05 imply that the perturbation .k/ condition P16 holds. It is represented by the asymptotic expansion (6.3.37) that should .;k/ be taken for  D 0. Also, conditions T5 , P32 , and E005 imply that the perturbation .k/ condition P15 holds. It is represented by the asymptotic expansion (6.3.37) that should be taken for  D .0/ > 0.

6.3.4 Exponential expansions in mixed ergodic and large deviation theorems All results given in Chapter 3 can be applied to the regenerative processes ."/ .t /, t  0, that describe functioning of queueing M/G type systems with quick service and a bounded queue buffer. .k/ Conditions T5 , E05 , P32 correspond to a pseudo-stationary model with the absorption probabilities asymptotically vanishing to zero. Under these conditions, analogues of Theorems 3.4.1–3.4.3 and 3.4.6–3.4.8, which give exponential asymptotic expansions in the corresponding mixed ergodic and large deviation theorems, can be formulated.

408

6

Quasi-stationary phenomena in stochastic systems .0;k/

For example, if conditions T5 , E05 , the perturbation condition P32 , and the bal.r/ ancing condition B5 (for some 1  r  k) hold, the asymptotic relation given in Theorem 3.4.2 takes the following form for i; j ¤ 0: Pi f ."/ .t ."/ / D j; ."/ > t ."/ g ! ı.j; N /e r ar as " ! 0: expf.a1 " C    C ar 1 "r 1 /t ."/ g

(6.3.38)

If k  N  1 and the order condition O13 holds, then, as follows from Theorem 3.4.2, al D 0, l < N  1, while aN 1 > 0. In particular, condition O13 is implied by .k/ conditions T6 and P33 . ..0/ ;k/

Conditions T5 , E005 , and P32 correspond to a quasi-stationary model with the absorption probabilities asymptotically separated from zero. Under these conditions, analogues of Theorems 3.4.4–3.4.5 and 3.4.9–3.4.10, which give exponential asymptotic expansion in mixed ergodic and large deviation theorems, can be formulated. Also, analogues of Theorems 3.5.2–3.5.4 give higher order asymptotic expansions for quasi-stationary distributions. It is useful to note that the theorems listed above cover the pseudo-stationary case as well. The limit parameters, including the stationary and the quasi-stationary distributions, and the coefficients in the corresponding asymptotic expansions, can be calculated by using the corresponding algorithms and formulas given in Chapters 3–5. They should be specified for the imbedded semi-Markov processes ."/ .t /, t  0, and the regenerative processes ."/ .t /, t  0, with the use of the corresponding formulas given in Sections 6.2 and 6.3.

6.4 Quasi-stationary phenomena in birth-and-death type stochastic systems In this section, we investigate conditions for pseudo- and quasi-stationary asymptotics for stochastic systems birth-and-death type.

6.4.1 Semi-Markov processes of birth-and-death type Let ."/ .t /, t  0, for every "  0, be a semi-Markov process with the phase space X D f0; 1; : : : ; N g and the transition probabilities 8 ."/ ."/ ˆ pi;C Fi .t / ˆ ˆ ˆ < p ."/ F ."/ .t / ."/ i; i Qij .t / D ˆ 1 ˆ ˆ ˆ :0

if i D 1; : : : ; N  1, j D i C 1 or i D j D N , if i D 1; : : : ; N  1, j D i  1, if i D j D 0, otherwise.

(6.4.1)

6.4

Quasi-stationary phenomena in birth-and-death type stochastic systems

409

."/

The imbedded discrete time Markov chain n , n D 0; 1; : : :, for the semi-Markov process ."/ .t /, t  0, has the matrix of transition probabilities, 3 2 1 0 0 ::: 0 0 0 7 6 p ."/ 0 p ."/ : : : 0 0 0 7 6 1; 1;C 7 6 :: :: :: :: :: :: :: 7: P."/ D 6 (6.4.2) : : : : : : 7 6 : 7 6 ."/ ."/ 0 0 : : : pN 1; 0 pN 1;C 5 4 0 ."/ ."/ 0 0 0 ::: 0 pN; pN;C ."/

."/

As in Chapter 4, we denote by j and j the first hitting time to a state j 2 X for ."/

the imbedded Markov chain n , n D 0; 1; : : :, and the semi-Markov process ."/ .t /, t  0, respectively. The state 0 is an absorption state and, therefore, the random variable ."/ 0 is the absorption time that is an object of our study. The condition T1 takes in this case the following form: T7 :

."/ .0/ (a) pi;˙ ! pi;˙ as " ! 0, i ¤ 0;

(b) Fi."/ ./ ) Fi.0/ ./ as " ! 0, i ¤ 0. Condition E2 becomes: .0/

E7 : pi;C > 0, i ¤ 0. Condition E7 can be realised in two variants. Let us introduce the set .0/

E D f1  i  N W pi; D 0g: The following condition is the first variant for realisation of condition E7 : .0/

E07 : pi;C > 0, i D 1; : : : ; N , and E ¤ ¿. In this case, denote .0/ iC D max.i  1 W pi; D 0/;

.0/ i D min.i  1 W pi; D 0/:

This is a pseudo-stationary case. The hitting probabilities satisfy 0 fij.0/ D 1 for

i  i , j  i , while 0 < 0 fij.0/ < 1 for 1  i < i , j  i . Also, the absorption .0/

.0/

probabilities satisfy jfi0 D 0 for i  i , j  i , while 0 < jfi0 < 1 for 1  i < i , .0/ j  i . The set of recurrent-without-absorption states is X1 D fiC ; : : : ; N g, while .0/ for the set of non-recurrent-without-absorption states, we have X2 D f1; : : : ; iC 1g. ."/ Conditions T7 and E07 do not guarantee that pi; D 0, i 2 E for " small enough. In order to have this property, one should require, additionally to T7 and E07 , the following

410

6

Quasi-stationary phenomena in stochastic systems

condition to hold: ."/

E8 : There exists a state i  2 E and "1 > 0 such that pi; > 0, i D i  C 1; : : : ; N , and pi."/  ; D 0 for every 0 < "  "1 .

Let conditions T7 , E07 , and E8 hold. Conditions T7 and E07 imply that there exists ."/ ."/ "2 > 0 such that pi;C > 0, i D 1; : : : ; N , and pi; > 0, i … E, i ¤ 0, for every "  "2 . Then, for 0 < "  "1 ^ "2 , the set of recurrent-without-absorption states is X1."/ D fi  ; : : : ; N g, while the set of non-recurrent-without-absorption states equals ."/ X2 D f1; : : : ; i   1g. .0/ Note also that conditions T7 , E07 , and E8 imply that pi; D 0, i.e., i  2 E. Thus, .0/

."/

i  i   iC and, therefore, X1 X1 for "  "1 ^ "2 . The following condition is the second variant for realisation of condition E7 : .0/ > 0, i D 1; : : : ; m, and E D ¿. E007 : qi;C

This is the quasi-stationary case. The hitting probabilities satisfy 0 < .0/ jfi0

.0/ 0 fij

0 such that pi;˙ > 0, i ¤ 0, for ."/

"  "3 . Then, the set of recurrent-without-absorption states is X1 D f1 : : : ; N g, ."/ while for the set of non-recurrent-without-absorption states, X2 D ¿, for 0 < "  "3 . Condition I4 takes in this case the following form: .0/

.0/

I7 : Fi .0/ < 1 for some state i 2 X1 . The non-periodicity conditions N1 and N2 have the same formulations as for the general semi-Markov processes. As was mentioned in Subsection 4.3.1, these conditions are not restrictive. For example, condition N1 holds if the distribution function .0/ Fi .t / has an absolutely continuous component at least for some i ¤ 0. In particular, this is the case for the standard birth-and-death type model. Note that, in this case, condition I6 holds as well.

6.4.2 Asymptotics of hitting and absorption probabilities Note first of all that condition T8 (a) implies that, for l ¤ 0, i; j 2 X, i ¤ j , ."/ iflj

.0/

! iflj

as " ! 0:

(6.4.3)

6.4

411

Quasi-stationary phenomena in birth-and-death type stochastic systems ."/

Let us consider, together with the usual hitting probabilities iflj

."/ i g,

.0/ i glj ,

."/

D Pl fj


0 such that > 0, r ¤ 0, for all "  "0 . Taking into account relation (6.4.13) we can write the following system of equations for the moment generating functions 0 r."/ r C1./, r D 1; : : : ; N  1, for every   0 and "  "0 : ."/ pr;˙

."/ 0 12 ./

D p1;C

."/

."/ 1 ./;

."/ 0 r r C1 ./

D pr;C

."/

."/ r ./

."/ C pr;

."/ ."/ ."/ r ./ 0 r1r ./ 0 r r C1./;

1 < r  N  1:

(6.4.14)

Note that the equations in (6.4.14) can take the form 1 D 1 if the corresponding moment generation functions take the value 1. Let us assume that 0 < ˇ < ı, where ı is taken from condition C19 , and "  "0 . ."/ The equations (6.4.14) yield that (a) 0 12 .ˇ/ < 1 and that (b) 0 r."/ r C1.ˇ/ < 1

."/ ."/ ."/ if and only if pr; r .ˇ/ 0 r1r .ˇ/ < 1 for given 1 < r  N  1. Also, equations ."/ ."/ ."/ (6.4.14) yield that (c) if pr; r .ˇ/ 0 r1r .ˇ/  1 and, therefore, 0 r."/ r C1 .ˇ/ D 1, ."/ ."/ hence 0 kkC1 .ˇ/ D 1 for every r  k  N 1, and (d) if pr; ."/ 0 r r C1.ˇ/

."/ ."/ r .ˇ/ 0 r 1r .ˇ/


1 if and only if pr;C r .ˇ/ C pr; r .ˇ/ 0 r 1r .ˇ/ >

414

6

Quasi-stationary phenomena in stochastic systems ."/

1 for given 1 < r  N  1. Also, equations (6.4.14) yield that (h) if 0 r 1r .ˇ/ > 1, ."/ then 0 kkC1 .ˇ/ > 1 for every r  k  N  1, while relation (6.4.12) implies that (i) ."/

."/

if 0 N 1N .ˇ/ > 1, then 0 N N .ˇ/ > 1. Consider the condition: C20 : There exist 1 < r  N  1 and 0 < ˇ < ı, for ı defined in condition C19 , .0/ .0/ .0/ .0/ .0/ .0/ .0/ such that: (a) pr; r .ˇ/ 0 r1r .ˇ/ < 1, and (b) pr;C r .ˇ/ C pr; r .ˇ/ .0/

 0 r 1r .ˇ/ > 1. .0/

The quantities 0 r1r .ˇ/, 1 < r  N  1, that enter condition C20 should be calculated recursively with the use of relations (6.4.15). The remarks above imply that the conditions C19 and C20 are sufficient for condition C13 to hold. Another variant of conditions that in quasi-stationary case are sufficient for condition C13 to hold are the corresponding variants of conditions G1 and G2 . For i ¤ 0, denote i D sup.  0 W

.0/ i ./

< 1/;

 D min.i ; i ¤ 0/:

Conditions G1 and G2 take in this case the following form: G3 : .a/  > 0; .b/

.0/  i . /

D 1 for some i ¤ 0;

and G4 : lim"!0

."/ i ./

< 1, i ¤ 0, for  <  .

Conditions T7 , E007 and C19 –C20 or G3 –G4 also imply that conditions C14 and C15 hold. Condition C16 is not needed, since, under conditions T7 and E007 , the set of non.0/ recurrent-without absorption states is X2 D ¿.

6.4.4 Mixed ergodic and limit/large deviation theorems Conditions T7 , and E07 correspond to the pseudo-stationary model with the absorption probabilities asymptotically vanishing to zero. Under these conditions, analogues for mixed ergodic and limit Theorems 4.4.1 and 4.4.2 and mixed ergodic and large deviation Theorems 4.6.1 and 4.6.2 covering the case of the pseudo-stationary asymptotics can be formulated. Also, the analogues of Theorems 4.4.3 and 4.4.5 give convergence conditions for the corresponding stationary distributions under the additional non-absorption condition A10 . Conditions T7 , and E007 correspond to a quasi-stationary model with the absorption probabilities asymptotically separated from zero. Under these conditions, analogues

6.4

Quasi-stationary phenomena in birth-and-death type stochastic systems

415

of mixed ergodic and large deviation Theorems 4.6.3, 4.6.5, 4.6.8, 4.6.7, and 4.6.11 covering the case of quasi-stationary asymptotics can be formulated as well as Theorem 4.6.6. They give conditions for convergence of the corresponding quasi-stationary distributions. Note that the theorems listed above can also be applied in the pseudostationary case as well. The limit parameters, including the stationary and quasi-stationary distributions can be calculated by using the corresponding algorithms and formulas given in Chapter 4. They should be specified for the semi-Markov processes of birth-and-death type ."/ .t /, t  0, introduced in this section.

6.4.5 Perturbation conditions in mixed ergodic and large deviation theorems Let us formulate perturbation conditions, which replace the perturbation conditions introduced in Chapter 5 in the theorems representing exponential asymptotic expansions in mixed ergodic and limit/large deviation theorems. We shall assume that conditions T7 , E7 , and C19 hold. .k/ The perturbation condition P18 take in this case the following form: .k/ ."/ .0/ : pi;˙ D pi;˙ Cei;˙ Œ1"C  Cei;˙ Œk"k Co."k / for i ¤ 0, where jei;˙ Œlj < 1, P34 i ¤ 0. .0/ It is convenient to define ei;˙ Œ0 D pi;˙ for i ¤ 0. .k/

Note that condition P34 (for any k D 0; 1; : : :) implies condition T7 . For i ¤ 0 and   0, denote ."/ i Œ; n

Z D

0

1

t n e t Fi."/ .dt /:

Condition C19 guarantees that for every 0 < ˇ < ı there exists "0 D ".ˇ/ > 0 such ."/ that for n D 0; 1; : : :, i ¤ 0, and "  "0 , (a) i Œ; n < 1. .;k/

The perturbation condition P26 .;k/ : P35

."/ i Œ; n

D n D 0; : : : ; k, i ¤ 0.

takes in this case the following form for  < ˇ:

.0/ kn C i Œ; nCei Œ; 1; n"C  Cei Œ; k n; n" i ¤ 0, where jen Œ; r; nj < 1, r D 1; : : : ; k  n, n

It is convenient to define ei Œ; 0; n D .;k/ P35

.0/ i Œ; n,

o."kn / for D 0; : : : ; k,

for n D 0; : : : ; k, i ¤ 0.

Note that condition should be used for the value  D 0 in the pseudostationary case, where conditions T7 , E07 , and C19 hold. In this case, i."/ Œ0; n D R 1 n ."/ 0 t Fi .dt /, n D 1; : : :, are, in fact, power moments for the distribution functions ."/ Fi .t /.

416

6

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.;k/

Condition P35 should be used for the value  D .0/ > 0, where .0/ is a unique ."/ root of the characteristic equation 0 N N ./ D 1 in the quasi-stationary case, where conditions T7 , E007 , and C19 hold.

6.4.6 Pivotal properties of asymptotic expansions for absorption probabilities .k/ According to Lemma 5.1.5, condition P34 implies that the absorption probabilities have the following asymptotic expansion for i ¤ 0: ."/ Nfi0

.0/

D Nfi0 C N bi0 Œ1" C    C N bi0 Œk"k C o."k /:

(6.4.16)

.k/

In the quasi-stationary case, i.e., if conditions P34 and E007 hold, the limit absorption probabilities satisfy Nfi0.0/ > 0, i ¤ 0, and, therefore, the asymptotic expansion (6.4.16) is .0; k/-pivotal. Let us consider in more details a more interesting pseudo-stationary case, where .k/ conditions P34 and E07 hold. We restrict consideration to the most important case, where the following asymp.k/ : totic order condition holds in addition to conditions P34 O19 : ei; Œ0 D 0; ei; Œ1 > 0, i ¤ 0. ."/ D O."/, i ¤ 0, and E D In this case, the transition probabilities satisfy pi f1; : : : ; N g. Formula (6.4.9) implies, in this case, that, for 1  i < N ,

."/ Nfi0

D 1



i Y r D1

."/ Ai Œ0; N  ."/ AN Œ0; N  ."/ pr; 

PN 1 Ql D

i Y

."/

pr; rD1 p ."/ r;C

lDi

."/

AN Œ0; N  ! er; Œ1 "i as " ! 0:

(6.4.17)

r D1

Asymptotic relation (6.4.11) yields, since E D f1; : : : ; N g, that there is the pivotal ."/ asymptote given by relation (6.4.17) for the cyclic absorption probabilities NfN 0 as well, i.e., ."/ NfN 0

."/ QN pr;   ."/ ."/ rD1 A 1 Œ0; N  p ."/ ."/ D pN; 1  N."/ D pN;C ."/ r;C AN Œ0; N  AN Œ0; N  ! N Y Y ."/  pr;  er; Œ1 "N as " ! 0:

r 2E

r D1

(6.4.18)

6.4

Quasi-stationary phenomena in birth-and-death type stochastic systems

417

Let us now consider a more general case, where the following asymptotic order .k/ : condition holds additionally to conditions E07 and P34 O20 : ei; Œl D 0, l < ki , ei; Œki  > 0 for i 2 E 0 E, where 1  ki  k, i ¤ E 0 and ei; Œl D 0, l  k, for i 2 E 00 D E n E 0 . ."/ ."/ D O."ki /, i 2 E 0 , pi; D o."k /, In this case, the transition probabilities are pi; ."/

.0/

i 2 E 00 , while pi; ! pi; > 0, i … E. P P Let us also define j.i / D min.j 2 E; j > i /, hi D r2E 0 ;r i kr ; h D r2E 0 kr , where hQ i is the number of states in the set E 00 \ fj ¤ 0; j  i g and hQ is the number of states in E 00 . In this case, the absorption probabilities have the following asymptotics as " ! 0: PN 1 Ql ."/ Nfi0

lDi

."/ pr; r D1 p ."/ r;C

 A."/ Œ0; N  N 8   .0/ Pj.i/1 Q Q ˆ pr; ."/ ˆ if 1  i < iC , ˆ r 2E;r i pr;  r …E;r l p .0/ ˆ lDi ˆ ˆ  r;C  n  1; n1 D j /:

nD1 ."/

The case N D 1 is trivial, since 1 u11 D 1. Let N > 1. If j D N then, for i ¤ 0, ."/ N uiN

D ı.N; i /:

(6.4.27)

Consider the case where 1  j < N . The distributions of the random variable  j < N , can be expressed in terms of the hitting probabilities introduced in formulas (6.4.4) and (6.4.9). They take the following form for n D 1; 2; : : :: 8 ."/ ."/ ˆ g .g /n1 if 1  i < j < N , ˆ ˆ 0 ij j ˆ < .g ."/ /n1 ; if 1  i D j < N , ."/ j Pi fıNj  ng D (6.4.28) ."/ ."/ n1 ˆ if 1  j < i < N , ˆ .1  j giN /.gj / ˆ ˆ ."/ ."/ : ."/ pN; .1  j gN 1N /.gj /n1 if 1  j < i D N , ."/ ;1 ıNj

where, ."/

gj

."/

."/

."/

."/

D pj; 0 gj 1j C pj;C .1  j gj C1N /:

It follows from formula (6.4.28) that, for 1  j < N , 8 ."/ ."/ ˆ if 1  i  j 0 gij =.1  gj / ˆ ˆ ˆ < 1=.1  g ."/ / if 1  i D j ."/ ."/ j N uij D Ei ıNj D ."/ ."/ ˆ .1  j giN /=.1  gj / if 1  j < i ˆ ˆ ˆ ."/ ."/ : ."/ pN; .1  j gN 1N /=.1  gj / if 1  j < i

(6.4.29)

< N, < N,  N,

(6.4.30)

D N. .k/

Let us restrict the considerations to the basic case where conditions P34 and O19 hold. Œi; j  ! 1 as " ! 0, while (b) Formula (6.4.9) implies that, for i < k  j , (a) A."/ k Qk ."/ ."/ ki Aj Œi; j   Ak Œi; j   . r DiC1 er; Œ1/" as " ! 0, and, therefore, ."/ i gkj

."/

."/

D Ak Œi; j =Aj Œi; j  ! 1 as " ! 0;

(6.4.31)

420

6

Quasi-stationary phenomena in stochastic systems

and ."/

."/

."/

."/

1  i gkj D .Aj Œi; j   Ak Œi; j /=Aj Œi; j  

 Y k

 er; Œ1 "ki as " ! 0:

(6.4.32)

r DiC1

Relations (6.4.31) and (6.4.32) imply that ( .ej; Œ1 C ej; Œ1/" C o."/ ."/ gj D e2; Œ1" C o."/

if 1 < j < N , if j D 1.

(6.4.33)

Finally, using formula (6.4.30) and the asymptotic relations (6.4.31), (6.4.32) and (6.4.33) and applying the proposition (iii) from Lemma 8.1.1, we get the following asymptotic relation, which gives a complete description of pivotal properties of the potential matrices: ."/ N uij

8 ˆ 1 C e2; Œ1" C o."/ ˆ ˆ ˆ ˆ ˆ ˆ ˆ 1 C .ej; Œ1 C ej C1; Œ1  e1; Œ1/" C o."/ ˆ < 1 C .e Œ1 C e j; j C1; Œ1/" C o."/ D ˆ 0 ˆ ˆ ˆ ˆ ˆ 1 ˆ ˆ ˆ : .Qi ij C o."ij / r Dj C1 er; Œ1/"

if 1 D i D j , if 1 D i < j < N , if 1 < i  j < N , if 1  i < j D N , if i D j D N , if 1  j < i  N .

(6.4.34)

The asymptotic relation (6.4.34) permits, in particular, to give explicit values for elements of the first two matrices in the asymptotic expansion (6.4.24) for potential matrices, 8 ˆ 1 if 1  i  j < N , ˆ ˆ ˆ < 0 if 1  i < j D N , .0/ (6.4.35) N uij D N uij Œ0 D ˆ 1 if i D j D N , ˆ ˆ ˆ : 0 if 1  j < i  N , and

8 ˆ ˆ ˆ e2; Œ1 ˆ ˆ ˆ ˆ ˆ ej; Œ1 C ej C1; Œ1  e1; Œ1 ˆ < e Œ1 C e j; j C1; Œ1 u Œ1 D N ij ˆ 0 ˆ ˆ ˆ ˆ ˆ ej C1; Œ1 ˆ ˆ ˆ :0

if 1 D i D j , if 1 D i < j < N , if 1 < i  j < N , (6.4.36) if 1  i  j D N , if 1  j D i  1  N  1, if 1  j < i  1  N  1.

6.4

Quasi-stationary phenomena in birth-and-death type stochastic systems

421

Two cases should be considered, (1) k  N or (2) k < N . .k/ In case (1), if conditions P34 , E07 , and O19 hold, the asymptotic relation (6.4.34) .0/ implies that Z0 D X1 D fN g, Zr D fN  rg, r D 1; : : : ; N  1, and Zr D ¿, r D N; : : : ; k C 1 (these sets were defined in Subsection 5.1.2). Also, Y0;0 D ¿, Y0;1 D f1g, Y0;r D ¿; r D 1; : : : ; k, and Y0;kC1 D f2; : : : ; N g (these sets were defined in Subsection 5.1.5). Consequently, W0;r D ¿, r D 0; : : : ; N  1, W0;N D .N/ f1g, W0;r D ¿, r D N C 1; : : : ; k, and W0;kC1 D f2; : : : ; N g. Condition O4 holds in this case. ."/ By Theorem 5.4.1, the characteristic root ."/ of the equation 0 N N ./ D 1 has, in this case, the following .N; k/-asymptotic pivotal expansion: ."/ D aN "N C    C ak "k C o."k /:

(6.4.37)

.k/ , E007 , and O19 hold, then the asymptotic relation In case (2), if conditions P34 (6.4.34) implies that Z0 D X1.0/ D fN g, Zr D fN  rg, r D 1; : : : ; k, and ZkC1 D f1; : : : ; N  k  1g (these sets were defined in Subsection 5.1.2). Also Y0;0 D ¿, Y0;1 D f1g, Y0;r D ¿, r D 1; : : : ; k, and Y0;kC1 D f2; : : : ; N g (these sets were defined in Subsection 5.1.5). Hence, W0;r D ¿, r D 0; : : : ; k, .k/ W0;kC1 D f1; : : : ; N g. Condition O3 holds in this case. By Theorem 5.4.1, the ."/ characteristic root of the equation 0 N N ./ D 1 has the asymptotics ."/ D o."k /. .k/

The case, where conditions P34 and O20 hold, can be treated in an analogous way. The pivotal properties of the asymptotic expansion for the characteristic root ."/ can be analysed with the use of the asymptotic relation (6.4.19). Indeed, ."/ has a ."/ pivotal .h; k/-expansion for some h  k if the cyclic absorption probabilities NfN 0 ."/ have pivotal .h; k/-expansions, and ."/ D o."h / if NfN 0 D o."h / for some h  k. Thus, if the parameters h and hQ that enter the asymptotic expansion (6.4.19) take values, respectively, less than or equal k and equal to 0, then ."/ has a pivotal .h; k/Q expansion; if h C hQ  k and hQ > 0, then ."/ D o."hCh /; and if h C hQ > k, then ."/ D o."k /.

6.4.8 Standard birth-and-death type processes A standard birth-and-death type model corresponds to the case where the distribution functions Fi."/ .t / D 1  expfqi."/ t g, t  0, i ¤ 0, are exponential with parameters ."/ qi > 0 for i ¤ 0. In the case of a standard birth-and-death type model, condition T7 takes the following form: T8 :

."/

.0/

(a) pi;˙ ! pi;˙ as " ! 0, i ¤ 0; ."/

(b) qi

.0/

! qi

> 0 as " ! 0, i ¤ 0.

422

6

Quasi-stationary phenomena in stochastic systems

In this case, as follows from Lemma 4.5.10, conditions T8 and E007 imply that con.0/ ditions G3 and G4 automatically hold with  D min.qi ; i ¤ 0/. In the case of a standard birth-and-death type model mentioned above, the pertur.;k/ can be replaced with the following condition: bation condition P35 .k/

."/

.0/

P36 : qi D qi C qi Œ1" C    C qi Œk"k C o."k / for i ¤ 0, where jqi Œrj < 1 for i ¤ 0 and r D 1; : : : ; k. .0/

It is convenient to define qi Œ0 D qi , for i ¤ 0. In this case the corresponding power moments and the moment generating functions are given by the following formulas for i ¤ 0 and n D 0; 1; : : :: Z 1 ."/ ."/ ."/ ."/ s n qi expfqi sg ds D nŠ.qi /n < 1; (6.4.38) mi Œn D 0

and ."/ i Œ; n

Z D

1 0

s n e s qi."/ expfqi."/ sg ds ."/

D

nŠqi ."/ .qi



/nC1

< 1;

."/

 < qi :

(6.4.39)

These expansions can be obtained with the use of Lemmas 5.6.3 and 5.6.5. Note .k/ that the only condition P36 is required. ."/ ."/ In applications, transition probabilities pi;˙ , i ¤ 0, and the parameters qi , i ¤ 0,

."/ ."/ are expressed in terms of the corresponding transition intensities qi;˙ D giN;˙ q˙ , i ¤ 0, by the following formulas: ."/ ."/ qi."/ D giN; q C giN;C qC ;

."/ ."/ ."/ pi;˙ D giN;˙ q˙ =qi ;

(6.4.40)

."/

where (a) giN;˙ > 0, i ¤ 0, and (b) q˙ > 0, for " > 0. In the case of a standard birth-and-death type model, condition T8 takes the following form: ."/

.0/

T9 : q˙ ! q˙ > 0 as " ! 0. Conditions E07 and E007 that correspond, respectively, to pseudo- and quasi-stationary cases take the following form: .0/

.0/ D 0; E09 : qC > 0, q

and .0/

E009 : q˙ > 0.

6.4

Quasi-stationary phenomena in birth-and-death type stochastic systems .k/

423

.k/

In this case, the perturbation conditions P34 and P36 can be replaced with the following condition: .k/

."/

.0/

P37 : q˙ D q˙ C q˙ Œ1" C    C q˙ Œk"k C o."k /, where jq˙ Œrj < 1, for r D 1; : : : ; k. .0/

It is convenient to define q˙ Œ0 D q˙ . .k/

.k/

.k/

Condition P37 implies, by Lemma 8.1.1, that both conditions P36 and P34 hold with the coefficients in the corresponding asymptotic expansions given for r D 0; : : : ; k and i ¤ 0 by qi Œr D giN; q Œr C giN;C qC Œr;

(6.4.41)

and   rX 1 pi;˙ Œmqi Œr  m =qi Œ0: pi;˙ Œr D giN;˙ q˙ Œr 

(6.4.42)

mD0

6.4.9 Exponential expansions in mixed ergodic and large deviation theorems All results given in Chapter 5 can be used for semi-Markov processes of birth-anddeath type. .k/ .k/ Conditions T7 , E07 , and P34 –P37 correspond to a pseudo-stationary model with the absorption probabilities asymptotically vanishing to zero. Under these conditions, analogues of Theorems 5.4.1, 5.4.2, which give exponential asymptotic expansion in mixed ergodic and large deviation theorems, can be formulated. Also an analogue of Theorem 5.5.1 gives higher order asymptotic expansions for stationary distributions under the additional non-absorption condition A8 . .k/ .k/ Conditions T7 , E007 , and P34 –P37 correspond to a quasi-stationary model with the absorption probabilities asymptotically separated from zero. Under these conditions, analogues of Theorem 5.4.3 and 5.4.4, which give an exponential asymptotic expansion in mixed ergodic and large deviation theorems, can be formulated. Also, analogues of Theorems 5.5.4 and 5.5.5 give higher order asymptotic expansions for quasistationary distributions. Note that these theorems can also be applied in the pseudostationary case. The limit parameters, including the stationary and the quasi-stationary distributions, and the coefficients in the corresponding asymptotic expansions, can be calculated by using the corresponding algorithms and formulas given in Chapters 4 and 5. They are used for the semi-Markov processes ."/ .t /, t  0, introduced in this section.

424

6

Quasi-stationary phenomena in stochastic systems

6.4.10 Nonlinearly perturbed queueing systems of M/M type There are many models of queueing systems of M/M type that can be described with the use of standard birth-and-death type Markov processes. For example, let us consider a standard birth-and-death type Markov process with transition intensities given by the following formula for i ¤ 0: ."/

."/ ; qi; D i q

."/

."/

qi;C D .N  i /qC :

(6.4.43)

A queueing system with N servers working independently and possessing expo."/ and q ."/ , respectively, can nential working and repairing times with intensities q C be described with the use of a continuous time Markov chain ."/ .t /, t  0, with transition intensities given by formula (6.4.43). In this case, ."/ .t / is the number of working servers at the moment t , while the lifetime is the first moment at which all the servers are being repaired, i.e., the process ."/ .t / occurs in the state 0. According to the definition given above, the intensity of the virtual transition N ! N given by formula (6.4.43) equals 0 and, consequently, the corresponding transition ."/ D 0. This contradicts the conditions E07 and E007 , according to probability is pN;C ."/

which the probabilities pN;C should take positive values for all " small enough. This side effect can be easily overcomed by applying the regularisation procedure described ."/ ."/ D 0 can be replaced with a new value qQ N;C D in Subsection 5.6.2. The intensity qN;C q > 0. As was explained in Subsection 5.6.2, this regularisation procedure does affect neither the Markov chain ."/ .t / nor the hitting time to a give state. In queueing terms, the additional virtual transitions N ! N can be interpreted as a realisation of special test procedures which confirm that all servers are in the working position and do not require repairing. Another example is a standard birth-and-death type Markov process with transition intensities given by the following formula for i ¤ 0: ."/ ."/ D i q ; qi;

."/ ."/ qi;C D qC :

(6.4.44)

These intensities also correspond to a model of queueing system with N servers working independently and possessing exponential working and repairing times with ."/ ."/ the intensities q and qC , respectively. The difference with the former one is that there is only one service place for repairs in the system. The broken servers can be repaired only in a sequential order while, in the former model, simultaneous repairs of all broken servers were possible. As in the first example, ."/ .t / is the number of working servers at the moment t , while the lifetime is the first moment at which all servers occur in a repairing state, i.e., the process ."/ .t / occurs in the state 0. The regularisation procedure is not required for the model described above.

6.4

Quasi-stationary phenomena in birth-and-death type stochastic systems

425

Many other modification of queueing systems of M/M type can be considered with the use of standard birth-and-death type Markov processes. The results presented in the current section permit to make an asymptotic analysis of pseudo- and quasi-stationary phenomena in the corresponding queueing systems with nonlinearly perturbed parameters.

6.4.11 Quasi-stationary phenomena in population and epidemic models As is well known, standard birth-and-death type Markov processes can be also used for a description of various types of population and epidemic models. In particular, the classical population model is described by the birth-and-death process with transition intensities given by the following formula, for i ¤ 0: ."/

."/

."/ qi; D i q ;

."/

qi;C D i qC :

(6.4.45)

."/

."/ are, respectively, the individual birth and the death intensities. Here, qi;C and q ."/ In this case,  .t / is the number of individuals in the population at the moment t , while the lifetime is the first moment at which individuals die, i.e., the process ."/ .t / occurs in the state 0. Note that we consider a model with a bounded maximal size N of population. The regularisation procedure is not required for the model described above. Another classical example is an epidemic model described by a birth-and-death process with the transition intensities given by the following formula for i ¤ 0: ."/

."/ qi; D i q ; ."/

."/

."/

qi;C D i.N  i /qC :

(6.4.46)

."/

Here qi; and qC are, respectively, individual recovering and contamination intensities. In this case, ."/ .t / is the number of ill individuals in a population at the moment t , while the epidemic lifetime is the first moment when all individuals recover, i.e., the process ."/ .t / occurs in the state 0. Note that the model with bounded maximal size of population N is considered. According to the definition given above, the intensity of the virtual transition N ! N given by formula (6.4.46) equals 0 and, consequently, the corresponding transition ."/ probability is zero, pN;C D 0. This contradicts conditions E07 and E007 , according to ."/

which the probabilities pN;C should take positive values for all " small enough. This side effect can be easily corrected by applying the regularisation procedure described ."/ ."/ in Subsection 5.6.2. The intensity qN;C D 0 can be replaced with a new value qQ N;C D q > 0. As was explained in Subsection 5.6.2, this regularisation procedure does not affect either the Markov chain ."/ .t / or the time hitting a give state. Many other modifications of population and epidemic models type can be considered with the use of standard birth-and-death type Markov processes. The results

426

6

Quasi-stationary phenomena in stochastic systems

given in the current section permit to make an asymptotic analysis of pseudo- and quasi-stationary phenomena in the corresponding population and epidemic models with nonlinearly perturbed parameters.

6.5 Quasi-stationary phenomena in metapopulation models Natural populations of most species have a spatial structure. Several habitat patches can support local populations. Such a set of local populations is called a metapopulation. Local populations in a metapopulation are connected by migration. A local population may go extinct due to a catastrophe. An empty patch may be colonised by migrants originating from other patches. It is useful to note that there is a clear analogy between metapopulation models and epidemic models and that, in fact, the epidemic models are particular cases of metapopulation models. The analogy is actually takes place at different hierarchical levels. The most obvious analogy identifies patches with host individuals, colonisation with infection, and local extinction with recovery. Occupied patches correspond to infected, and empty ones to susceptible individuals. Another interpretation is to identify patches with groups of individuals, for instance, households or other groups of individuals with similar behaviour.

6.5.1 Metapopulation models We consider a system consisting of m patches which at any time t  0 can be either occupied or empty. The state of the metapopulation at time t is described by the vector ."/ ."/ ."/ ."/ .t / D .1 .t /; : : : ; m .t //, t  0, where i .t / is either 1 or 0, according to th whether the i patch is occupied or empty at time t . Now let us describe a stochastic process, which defines the metapopulation model, more precisely. We suppose that (a) in the absence of migration, the local population inhibiting a patch i will go extinct in a time interval of length t with probabil."/ ."/ ity pi i .t / D gi i t C o.t /, (b) the probability that an empty patch j will be colonised in a time interval of the length t by migrants originating from patch i is ."/ ."/ pij .t / D gij t Co.t /, (c) all interaction processes (extinction and colonisation) are independent. The local dynamics is thus modelled by preassigning an m  m interaction matrix ."/ ."/ ."/ G D kgij k, where 0  gij < 1, i; j D 1; : : : ; m. Having described the dynamics of a local patch we can deduce the law governing the time evolution of the process ."/ .t /. We suppose that the process ."/ .t /, t  0 is a continuous from the right homogeneous Markov chain with absorption, which is a particular case of the model considered in Section 5.6. The phase space of the random process ."/ .t / is the m-dimensional binary hypercube X D fxN D .x1 ; : : : ; xm / W xi 2 f0; 1gg. The state 0N D .0; : : : ; 0/ corresponding

6.5

427

Quasi-stationary phenomena in metapopulation models ."/

to metapopulation extinction is an absorption state, and the first time 0N the process hits this state can be interpreted as the extinction time of the metapopulation. The local transition intensities are given by

."/ qxN yN

D

8 ."/ ˆ 0, i ¤ j . E10 : gij

Condition E10 can be realised in two variants. Condition T10 obviously implies that .0/ the limit intensities gi i  0, i D 1; : : : ; m. Let us introduce the set .0/

E D f1  i  m W gi i D 0g: The following condition is the “pseudo-stationary” variant for realisation of condition E10 : .0/

E010 : gij > 0, i ¤ j , and E ¤ ¿. In this case we assume, without loss of generality, that there exists 1  l  m such that E D f1; : : : ; lg. We shall refer to patches with the indices i 2 E as “mainland” patches. Indeed, in the limit case " D 0, the patches i 2 E act as mainland; its population will never go extinct. It is obvious that conditions T10 and E010 imply that the following condition holds: ."/

."/

A10 : gij > 0; i ¤ j , and gi i > 0, i D l C 1; : : : ; m, "  "1 for some "1 > 0. Condition A10 guarantees that, for every model with the parameter " small enough, every patch with index j D 1; : : : ; m can be colonised, i.e., the state 0 changes to the P ."/ state 1 with the positive colonisation intensity iWxi D1 gij , and that every patch with index i D l C 1; : : : ; m can change the state 1 to the state 0 with the positive extinction intensity gi."/ i . As far as patches with the indices i D 1; : : : ; l are concerned, it is only certain, due to conditions T10 and E010 , that the corresponding extinction intensities are small for " small enough and that these patches can not become empty for the unperturbed model (" D 0). As was mentioned above, conditions T10 and E010 do not guarantee that the extinc."/ tion intensities vanish, gi i D 0, i D 1; : : : ; l, for " small enough. In order to guar-

6.5

Quasi-stationary phenomena in metapopulation models

429

antee this property, one needs to impose, additionally to T10 and E010 , the following condition: ."/

A11 : gi i D 0, i 2 E, "  "2 for some "2 > 0. Condition A10 and A11 mean that, for every model with the parameter " small enough, every patch with index i D 1; : : : ; m can be colonised, every patch with index i D 1; : : : ; l can not become empty, while every patch with index i D l C 1; : : : ; m may become empty. The following condition is the “quasi-stationary” variant for realisation of condition E10 : .0/

E0010 : gij > 0, i ¤ j , and E D ¿. It is obvious that conditions T10 and E0010 imply that the following condition holds: ."/

A12 : gij > 0, i; j D 1; : : : ; m, "  "3 for some "3 > 0. Condition A12 means that, for every model with the parameter " small enough, every patch can become empty, i.e., a change from the state 1 to the state 0 is admitted, and every patch can be colonised, i.e., the processes can change the state 0 to the state 1. Gyllenberg and Silvestrov (1994) considered such a metapopulation model in discrete time. For the analysis of quasi-stationary phenomena of a similar model, but without the limit procedure, we refer to Day and Possingham (1995). The introduction for the introduction contains a more detailed explanation of the difference between quasi- and pseudo-stationary phenomena. .0/ N where 1N D .1; 1; : : : ; 1/. Condition E10 guarantees that qxN > 0 for every xN ¤ 1, P .0/ .0/ N The state 1 has a special status. For this state, the intensity satisfies q1N D m iD1 gi i .

.0/ D 0 and, therefore, If all the intensities are zeros, gi.0/ i D 0, i D 1; : : : ; m, then q1N N the state 1 is an absorption state. To change this situation, we shall use a regularisation transformation by adding an additional intensity q > 0 in this state for the virtual N transition 1N ! 1. ."/ ."/ ."/ Thus, let Q .t / D .Q 1 .t /; : : : ; Q m .t //, t  0, be for every "  0 a semi-Markov process with a phase space X and the transition probabilities

Qx."/ N yN .t / where

."/

qQ xN yN

."/  qQxN yN  D ."/ 1  expfqQ x."/ N tg ; qQxN

8 ˆ gi."/ ˆ i ˆ ˆ 0 for any x

6.5

Quasi-stationary phenomena in metapopulation models

431

Indeed, in this case, all colonisation and extinction intensities, which can provide a N are positive. transition from the state xN to the state yN without visiting the state 0, The property of hitting probabilities described above means that, for every 0  "  "1 , the Markov chain ."/ .t / has one class of recurrent-without-absorption states ."/ .0/ X1 X1 . As far as the absorption probabilities are concerned, condition A10 guarantees only ."/ that, for every 0  "  "1 , the absorption probabilities are positive, yN fxN 0N > 0, for .0/

any xN 2 Ul , yN 2 X1 . Indeed, in this P case, the class Ul consists of states of the form .0; : : : ; 0; xlC1 ; : : : ; xm / such that m kDlC1 ı.xk ; 1/  1. All colonisation and extinction intensities, which can provide any transition within the class Ul , are positive. The one-step transition to the absorption state 0N D .0; : : : ; 0/ is possible from the states in this class such that P m kDlC1 ı.xk ; 1/ D 1. Also, in the prelimit cases, i.e., if 0 < "  "1 , the absorption probabilities satisfy ."/ .0/ f N 2 X0 n Ul ; yN 2 X1 such that xi D 1 if the corresponding yN xN 0N > 0 for the states x ."/

extinction intensity is positive, gi i > 0, and xi D 0 if the corresponding failure ."/ intensity is gi i D 0. Indeed, in this case, it is possible that a transition to some state zN from the class Ul .0/ occurs without visiting the class X1 and then a transition from the state zN to the state 0N occurs without leaving the class Ul . The situation is different in the limit case where " D 0. In this case, as was pointed .0/ .0/ .0/ out above, the hitting probabilities satisfy zN fxN yN D 0 for any x; N zN 2 X1 , yN 2 X2 . This is so because all extinction intensities equal zero for patches with the indices i 2 E. .0/ Thus, in this case, X1 is a closed set of recurrent-without absorption states, while .0/ X2 is a class of non-recurrent-without absorption states. The absorption probabilities satisfy yN fxN.0/ D 0 for any xN 2 X0 n Ul , yN 2 X1.0/ but 0N

.0/ yN fxN 0N

2 .0; 1/, for xN 2 Ul , yN 2 X1.0/ . This also is true because all extinction intensities equal zero for the servers with indices i 2 E. Thus, the case where conditions T10 and E010 hold, corresponds to a pseudo-stationary model. Moreover, in this case, the class of recurrent-without-absorption states is .0/ .0/ X1 and the class of non-recurrent-without-absorption states is X2 . As was mentioned above, conditions T10 and E0010 imply condition A12 to hold. According to this condition, for every 0  "  "3 , the hitting probabilities are positive, ."/ N yN 2 X0 . This means that the Markov chain ."/ .t / has one 0N fxN yN > 0 for any x;

class of recurrent-without-absorption states X1."/ D X0 . Moreover, in this case, it can ."/ N yN 2 X0 . readily be seen that also yN fxN 0N > 0 for any x;

432

6

Quasi-stationary phenomena in stochastic systems

This holds because the colonisation and extinction intensities are positive for all patches. Thus, the case where conditions T10 and E0010 hold, corresponds to a quasi-stationary model. Moreover, in this case, the class of recurrent-without-absorption states is X1.0/ D X0 . Let us discuss how conditions introduced in Chapter 4 are realised for the model described above. We assume that conditions T10 and E10 hold. N yN 2 X, Relations (6.5.1), (6.5.2), and condition T10 obviously imply that, for x; ."/

.0/

."/

.0/

qxN yN ! qxN yN as " ! 0;

(6.5.8)

and qxN ! qxN

as " ! 0:

(6.5.9)

By formula (6.5.3), the distributions of sojourn times are exponential, for x; N yN 2 X, ."/ ."/ FxN."/ yN .t / D FxN .t / D 1  expfqxN t g;

t  0:

(6.5.10)

Relations (6.5.8) and (6.5.9), and formula (6.5.10) imply that condition T3 holds. Thus, condition T1 holds. Formulas (6.5.9) and (6.5.10) imply that condition I3 holds. Relation (6.5.9) and formula (6.5.10) also imply that conditions M11 and M12 hold. .0/ The non-arithmetic conditions N1 and N2 for return times in states xN 2 X1 obviously hold since the distributions of sojourn times are exponential. Let us define N  D min.qx.0/ N ¤ 0/: N ;x

(6.5.11)

By formula (6.5.5),  > 0. As was mentioned in Section 5.6, condition C12 holds for any 0 < ı <  . Conditions T10 and E010 correspond to the pseudo-stationary case. In this case, as follows from Lemma 4.5.8, condition C13 also holds. Conditions T10 and E0010 correspond to the quasi-stationary case. In this case, condition G1 holds with the parameter  given in formula (6.5.11). Condition G2 also .0/ holds since in this case X2 D ¿. Therefore, as follows from Lemma 4.5.10, conditions C12 and C13 also hold. Conditions C14 and C15 also hold as follows from the remarks made in Section 5.6. .0/ Condition C16 is not needed, since X2 D ¿ in the quasi-stationary case. So, all conditions introduced in Chapter 4 are verified. All results given in Chapter ."/ ."/ 4 can also be applied to the Markov chain ."/ .t / D .1 .t /; : : : ; m .t //, t  0 that describes stochastic dynamics of the metapopulation model introduced in the current section.

6.5

Quasi-stationary phenomena in metapopulation models

433

6.5.3 Mixed ergodic and limit/large deviation theorems Conditions T10 and E010 correspond to a pseudo-stationary model with the absorption probabilities asymptotically vanishing to zero. Under these conditions, we can obtain analogues of mixed ergodic and limit Theorems 4.4.1 and 4.4.2 and mixed ergodic and large deviation Theorems 4.6.1 and 4.6.2, covering the case of the pseudo-stationary asymptotics. Also, analogues of Theorems 4.4.3 and 4.4.5 give conditions for convergence of the corresponding stationary distributions under the additional non-absorption condition A11 . Conditions T10 and E0010 correspond to the quasi-stationary model with the absorption probabilities asymptotically separated from zero. Under these conditions, analogues of mixed ergodic and large deviation Theorems 4.6.3, 4.6.5, 4.6.7, and 4.6.11, covering the case quasi-stationary asymptotics can be formulated as well as Theorem 4.6.6, which give conditions for convergence of the corresponding quasi-stationary distributions. It is also useful to note that the theorems listed above can be applied in the pseudo-stationary case as well. The limit parameters, including stationary and quasi-stationary distributions, can be calculated by using the corresponding algorithms and formulas given in Chapter 4. They should be specified for the semi-Markov processes ."/ .t /, t  0, introduced in the current section.

6.5.4 Conditions for exponential expansions in mixed ergodic and large deviation theorems Let us formulate conditions that replace the corresponding perturbation conditions introduced in Chapter 5 in the corresponding theorems representing exponential asymptotic expansions in mixed ergodic and limit/large deviation theorems. .k/ The perturbation condition P30 can be replaced with the following condition: .k/ ."/ .0/ : gij D gij C gij Œ1" C    C gij Œk"k C o."k / for i; j D 1; : : : ; m, where P38 jgij Œrj < 1; r D 0; : : : ; k; i; j D 1; : : : ; m. .0/

It is also convenient to define gij Œ0 D gij ; i; j D 1; : : : ; m.

.k/ obviously implies condition T10 . Note that condition P38 .k/ .k/ Condition P38 also implies that condition P30 and the corresponding expansions ."/ ."/ N yN 2 X: for the transition intensities qxN yN and qxN take the following form for xN ¤ 0; ."/

.0/

qxN yN D qxN yN C qxN yN Œ1" C    C qxN yN Œk"k C o."k /; where, for r D 1; : : : ; k, 8 ˆ if jxN  yj N D 1; xi D 1; yi D 0; i i Œr 0, i 2 E. Condition O21 means that the extinction intensities for the main patches are of the order O."/. Let us first clarify pivotal properties for the cyclic potential quantities, ."/

."/ xN uxN yN

."/

xN ^0N

D ExN

X

nD1

."/

ı.n1 ; y/; N

.0/

xN 2 X1 ;

yN 2 X0 :

6.5

435

Quasi-stationary phenomena in metapopulation models

Two cases should be considered (1) k  l and (2) k < l. Recall that k  0 is the .k/ , while l  1 is order of the asymptotic expansion in the perturbation condition P38 the number of indices in the set E, i.e., the number of the main servers. ."/ As shown in Lemma 5.1.1, the potential quantities xN uxN yN admit .0; k/-expansions, ."/ xN uxN yN

D xN uxN yN Œ0 C xN uxN yN Œ1" C    C xN uxN yN Œk"k C o."k /:

(6.5.18)

.k/

Let us show that, (a) in case (1), where k  l, conditions E010 , P38 , and O21 imply .0/ that xN uxN yN Œn D 0, n < r, while xN uxN yN Œr > 0 for xN 2 X1 D Z0 , yN 2 Ur ; 0  r  l (the sets Ur were defined in relations (6.5.6) and (6.5.7)). According to Lemma 5.1.2, statement (a) is equivalent to that (b) the global asymptotic order is rO .x; N y/ N D r for states xN 2 U0 , yN 2 Ur , 0  r  l. Statement (b) follows from an obvious observation that (c) the local asymptotic order r.xN 0 ; xN 00 / equals 1 if xN 0 2 Un ; xN 00 2 UnC1 , for some n D 0; : : : ; l  1, or 0 otherwise. Let .xN 0 ; : : : ; xN q / be an irreducible .xN 0 ; xN 00 /-chain of states from X0 , where xN 0 D xN 0 2 U0 ; xN 00 D xN q 2 Ur . Obviously, (d) the minimal number of pairs .xN n ; xN nC1 / with the local asymptotic order r.xN n ; xN nC1 / D 1 in such an irreducible .xN 0 ; xN 00 /-chain is r. These pairs correspond to transitions from some state in the class Un to some state in the class UnC1 for n D 0; : : : ; r  1. This proves statement (b) and, consequently, statement (a). Note that the statement (a) implies that (e) Zr D Ur , r D 0; : : : ; l (the sets Zr were S defined in Section 5.1). Also, Zr D ¿, r D l C 1; : : : ; k C 1, since lnD0 Un D X0 . 0 Denote by Ul1 the set of states xN D .x1 ; : : : ; xm / such that xi D 1 for some 1  i  l and xj D 0 for j ¤ i , and by Ul00 the set of states xN D .x1 ; : : : ; xm / such that xi D 1 for some l C 1  i  m and xj D 0 for j ¤ i . By the definition, 0 Ul1

Ul1 and Ul00 Ul . ."/

Formula (6.5.3) and conditions E010 and O21 imply that (f) qxN 0N D 0 if xN 2

Sl2

nD0 Un ;

."/

0 (g) qxN 0N  gi; Œ1" as " ! 0 (note that gi; Œ1 > 0) if xi D 1; xN 2 Ul1 ,

0 ; (h) q ."/  g .0/ as " ! 0 (note that D 0 if xN 2 Ul1 n Ul1 for i D 1; : : : ; l, and qx."/ ii N 0N xN 0N

gi.0/ N 2 Ul00 , for i D l C 1; : : : ; m, and qx."/ D 0 if xN 2 Ul n Ul00 . i > 0) if xi D 1, x N 0N Using relations (f)–(h), the asymptotic expansions (6.5.12) and (6.5.14) and the quotient rule for the asymptotic expansions given in statement (iv) of Lemma 8.1.1, one can readily get properties similar to (f)–(h) for the one-step absorption transition probabilities px."/ D qx."/ =qx."/ N . These properties are: (i) the transition probabilities N 0N N 0N Sl2 ."/ ."/ .0/ are zero, pxN 0N D 0 if xN 2 nD0 Un ; (j) pxN 0N  gi; Œ1"=qxN as " ! 0 if xi D 1, ."/

."/

.0/

.0/

0 0 xN 2 Ul1 for i D 1; : : : ; l, and pxN 0N D 0 if xN 2 Ul1 n Ul1 ; (k) pxN 0N  gi; =qxN as

D 0 if xN 2 Ul n Ul00 . " ! 0 if xi D 1, xN 2 Ul00 for i D l C 1; : : : ; m, and px."/ N 0N Statements (i)–(k) imply that (l) the set Y0;r N (defined in Subsection 5.1.5) coincides 0 with Ul00 for r D 0, with Ul1 for r D 1, with ¿ for r D 2; : : : ; k, and with X0 n 0 .Ul1 [ Ul00 / for r D k C 1.

436

6

Quasi-stationary phenomena in stochastic systems

Statements (a) and (l) imply in an obvious way that (m) the set W0;r (defined in N 0 [ Ul00 for r D l, Subsection 5.1.5) coincides with ¿ for r D 0; : : : ; l  1, with Ul1 0 with ¿ for r D l C 1; : : : ; k, and with X0 n .Ul1 [ Ul00 / for r D k C 1. .k/

Therefore, (n) if conditions E010 and P38 , and O21 hold and the additional assump.l/ tion (1) k  l holds, then condition O4 holds. In this case, by Lemmas 5.1.5 and 5.1.7, the cyclic absorption probabilities ."/ f N 2 X1.0/ have .l; k/-asymptotic pivotal expansions (with the coefficients xN xN 0N , x N xN bxN 0N Œl > 0), ."/ xN fxN 0N

D xN bNxN 0N Œl"l C    C xN bNxN 0N Œk"k C o."k /:

(6.5.19) ."/

Also, by Theorem 5.4.1, the characteristic root ."/ of the equation 0N xN xN ./ D 1 has the following .l; k/-asymptotic pivotal expansion: ."/ D al "l C    C ak "k C o."k /:

(6.5.20) .k/

Similarly, it can be shown that (o) in case (2), where k < l, conditions E010 , P38 , and O21 imply that the coefficients satisfy xN uxN yN Œn D 0, n < r, while xN uxN yN Œr > 0, .0/ for xN 2 X1 D U0 , yN 2 Ur , 0  r  k (these sets were defined in relations (6.1.6) S .0/ and (6.1.7)), while (p) xN uxN yN Œn D 0; n < k C 1 for xN 2 X1 D U0 , yN 2 lrDkC1 Ur . S Statement (o) implies that (q) Zr D Ur , r D 0; : : : ; k, and ZkC1 D lrDkC1 Ur . The sets Y0;r N were introduced in statement (l). It follows from statements (o) and (l) that (r) the set W0;r coincides with ¿, for N r D 0; : : : ; k and W0;kC1 D X0 . N .k/

Therefore, (s) if conditions E010 and P38 , and O21 hold and the additional assump.k/ tion (1) k  l is true, then condition O3 holds. ."/ .0/ By Lemmas 5.1.5 and 5.1.7, the cyclic absorption probabilities xN fxN 0N , xN 2 X1 , ."/

.0/

satisfy the following relations: (t) xN fxN 0N D o."k /, xN 2 X1 . Also, by Theorem

5.4.1, the characteristic root ."/ of the equation 0N x."/ N xN ./ D 1 satisfies the relation (u) ."/ k  D o." /. The asymptotic relations and (t) and (u) are not surprising. One should not expect .k/ to get more precise asymptotics under the perturbation condition P38 that give exact asymptotics for transition intensities only up to the terms of the order "k . Let us also consider a more complicated case where the following condition, addi.k/ tional to conditions E010 and P38 , holds: O22 : gi i Œn D 0, n < ki , i 2 E and gi i Œki  > 0, i 2 E, for some 1  ki  k, i D 1; : : : ; m. Condition O22 means that the extinction intensities for the main patches i 2 E may be of the different orders O."ki /.

6.5

Quasi-stationary phenomena in metapopulation models

437

As above, two cases should be considered, (3) k  k1 C    C kl D kNl and (4) k < kNl . By repeating the asymptotic analysis carried out above, it can be shown that, under .k/ .kN / .k/ conditions E010 , P38 , and O22 , condition O3 l holds in case (3), or condition O4 holds in case (4). Conditions O5.;h/ , O.;h/ , as was mentioned in Subsection 5.3.7, 6 can not be expressed so explicitly in terms of coefficients in the asymptotic expansions .k/ used in the perturbation condition P38 . Here, we should use the coefficients of the ."/ .0/ corresponding expansions for the moment generating functions 0N xN xN ./, x 2 X1 , given in Lemma 5.3.5 and specified for the semi-Markov processes considered in the current section.

6.5.5 Exponential expansions in mixed ergodic and large deviation theorems All results given in Chapter 5 can be used for the semi-Markov processes ."/ .t /, t  0 that describe stochastic dynamics of metapopulation models. .k/ Conditions T10 , E010 , P38 , and O21 or O22 correspond to a pseudo-stationary model with the absorption probabilities asymptotically vanishing to zero. Under these conditions, there are analogues of Theorems 5.4.1 and 5.4.2, which give exponential asymptotic expansions in mixed ergodic and large deviation theorems. Also an analogue of Theorem 5.5.1 gives higher order asymptotic expansions for stationary distributions under the additional non-absorption condition A11 . .k/ correspond to a quasi-stationary model with the abConditions T10 , E0010 , and P38 sorption probabilities asymptotically separated from zero. Under these conditions, one can obtain analogues of Theorems 5.4.3 and 5.4.4, which give exponential asymptotic expansion in mixed ergodic and large deviation theorems. Also, analogues of Theorems 5.5.4 and 5.5.5 give higher order asymptotic expansions for quasi-stationary distributions. It is useful to note that the theorems listed above also cover the pseudo.k/ stationary case, i.e., they can also applied if conditions T10 , E010 , and P38 hold. The limit parameters, including the stationary and quasi-stationary distributions, and the coefficients in the corresponding asymptotic expansions, can be calculated by using the corresponding algorithms and formulas given in Chapters 4 and 5. They should be specified for the semi-Markov processes ."/ .t /, t  0, introduced in the current section.

Chapter 7

Nonlinearly perturbed risk processes

Chapter 7 contains results that extend the classical Cram´er–Lundberg and diffusion approximations for the ruin probabilities to a model of nonlinearly perturbed risk processes. Both approximations are presented in a unified way using the techniques of perturbed renewal equations developed in Chapters 1 and 2. Correction terms are obtained for the Cram´er–Lundberg and the diffusion type approximations, which provides a needed asymptotic behaviour of relative errors in the perturbed model. We study the dependence of these correction terms on the relations between the rate of perturbation and the rate of growth of the initial capital. We also give various variants of the diffusion type approximation, including the asymptotics for increments and derivatives of the ruin probabilities. In the traditional risk theory, the risk process u .t / D u C  t 

.t/ X

k ;

t  0;

(7.0.1)

kD1

is used to model functioning of an insurance company. In (7.0.1), u is a nonnegative constant that denotes an initial capital of a company, a positive constant  denotes the gross premium rate, .t / is the Poisson process (with a parameter ) that counts the number of claims to the company in the time-interval Œ0; t , k , k D 1; 2; : : :, is a sequence of nonnegative i.i.d. random variables (with a finite mean ) which are independent for the process .t /, and k is the amount of the k th claim. An important object of the study is the ruin probability for a company that has an initial capital u, P .u/ D Pf inf u .t / < 0g; t0

u  0:

Of a special interest is the asymptotics of the ruin probabilities for large values of the initial capital u. The loading rate of claims to the company is characterized by a constant ˛ D   . Only the subcritical case with ˛ < 1 is nontrivial (if ˛  1, then the ruin probability P .u/  1). The classical result, known as the Cram´er–Lundberg approximation, gives an asymptotics for the ruin probability under Cram´er type conditions conditions on the claim distribution G.t /, P .u/ !  as u ! 1; e u

(7.0.2)

440

7

Nonlinearly perturbed risk processes

where the constant , known as the Lundberg exponent, and the limit constant  are determined from the constant ˛ and the claim distribution G.u/. Of a special interest also is the asymptotics of the ruin probabilities for large values of u and the values of ˛ which are less than but close to 1. Here, the so-called diffusion approximation gives an answer. In the simplest case, it describes the asymptotic behaviour of the ruin probabilities when the initial capital is u ! 1, and, simultaneously, the growth premium rate satisfies  # . Under the condition that u ! 1 and ˛ ! 1 such that .1  ˛/u ! %, the diffusion approximation yields an asymptotics for the ruin probability, P .u/ ! e %=a1 as u ! 1;

(7.0.3)

where the constant a1 is determined by the claim distribution G.t /. Taking the asymptotic relations (7.0.2) and (7.0.3) as a starting point, we formulate the problem in a more general way. We consider a family u."/ .t /, t  0, of risk processes that depend on a small parameter "  0. The process u."/ .t / is considered as a perturbation of the process u.0/ .t / and, therefore, we assume some weak continuity conditions satisfied by the characteristic quantities  D  ."/ ,  D ."/ , and the distributions G.u/ D G ."/ .u/, considered as functions of ", at the point " D 0. Moreover, we allow for nonlinear perturbations, which means that the characteristic quantities of the perturbed risk processes are nonlinear functions of ". However, we restrict our attention to the case of a smooth perturbation and, hence, assume that these functions can be expanded in a power series with respect to " up to and including the order k. The object of our study is the asymptotic behaviour of the ruin probabilities P ."/ .u/ for u ! 1 and " ! 0. The relationship between the rates at which " tends to zero and the initial capital u tends to infinity has an influence upon the obtained results. Without loss of generality it can be assumed that u D u."/ is a function of the parameter ". The relationship between the rate of perturbation and the rate of growth of the initial capital is characterized by the following balancing condition: "r u."/ ! %r as " ! 0

(7.0.4)

for some positive integer r and %r 2 Œ0; 1/. Under the assumptions described above and some natural Cram´er type condition on the claim distributions G ."/ , we obtain the asymptotic relation P ."/ .u."/ / ! e %r ar  as " ! 0; expf. C a1 " C    C ar 1 "r 1 /u."/ g

(7.0.5)

where  and  are the same as in (7.0.2) and (7.0.3), and we give an explicit recurrence algorithm for calculating the coefficients a1 ; : : : ; ar as rational functions of the coefficients in the expansions for the characteristic quantities of the perturbed risk processes.

7.1 Cram´er–Lundberg and diffusion approximations

441

."/

The ruin probability P ."/ .u."/ / D Pfinf t0 0 .t / < u."/ g can be interpreted as ."/ a tail probability for the infimum of the risk process 0 .t /. With this interpretation, the asymptotic relation (7.0.5) takes the form of a large deviation theorem. Depending on whether %r equals zero or is positive, the balancing condition (7.0.4) states that either u."/ D o."r / or u."/ D O."r /. Following the standard terminology of large deviation theory we refer to this asymptotic behaviour of u."/ as different large deviation zones. The main new element in our results is a high order exponential asymptotic expansion which extends in a unified way both the classical Cram´er–Lundberg approximation and the diffusion approximation of ruin probabilities to a model of nonlinearly perturbed risk processes. The derived approximations have an optimal asymptotic behaviour of the relative errors. This behaviour depends on the balance between the order of the perturbation parameter " and the order of growth of the initial capital u. The approximations are supplemented with an explicit algorithm for calculating the asymptotic corrections. We pay an additional attention to the case of the diffusion approximation. In addition to asymptotics (7.0.5), we also give asymptotics for increments of the ruin probabilities and for densities of the non-ruin probabilities. Theorem 7.2.1 and Lemmas 7.2.1–7.2.3 give various diffusion type asymptotics for the ruin probabilities. Theorems 7.2.2 and 7.2.3 give asymptotics for increments and densities of ruin probabilities for nonlinearly perturbed risk type processes in the model of diffusion approximation. Theorem 7.3.1 gives an approximation for ruin probabilities that covers both Cram´er–Lundberg and diffusion approximations in the model where the gross premium rate, the parameter of the Poisson claim flow and the claim distribution are nonlinearly perturbed. Theorem 7.3.2 gives exponential expansions which improve the result of Theorem 7.3.1. Approximations for the ruin probabilities based on exponential asymptotic expansions of type (7.0.5) are given in Subsection 7.4.2. Theorems 7.5.1–7.5.3 give diffusion and Cram´er–Lundberg type approximations for distribution of the capital surplus prior and at ruin for nonlinearly perturbed risk processes. In conclusion, we would like to note that the results obtained in Chapter 7 also have interpretations in the queueing theory. As is known, the Cram´er–Lundberg approximation describes the asymptotics of steady tail probabilities for waiting times of the classical M/G/1 queueing system in the subcritical ergodic case, while the diffusion approximation describes the corresponding asymptotics in the critical so-called heavy traffic case. In this context, the results obtained in the present section yield new large deviation asymptotics for waiting times of perturbed M/G/1 queueing system.

7.1 Cram´er–Lundberg and diffusion approximations In this section we give a short summary of classical results concerning the Cram´er– Lundberg and the diffusion approximations for ruin probabilities.

442

7

Nonlinearly perturbed risk processes

7.1.1 Classical Cram´er–Lundberg approximation for ruin probabilities Let u .t /, t  0 be a standard risk process defined in the following way: u .t / D u C  t 

.t/ X

k ;

t  0;

(7.1.1)

kD1

where (a) u is a nonnegative constant, (b)  is a positive constant, (c) k , k D 1; 2; : : :, is a sequence of nonnegative i.i.d. random variables with a distribution function G.u/ that has a finite expectation  > 0, (d) .t /, t  0 is a Poisson process with parameter  > 0, (e) the sequence k , k D 1; 2; : : :, and the process .t /, t  0, are independent. The risk process (7.1.1) has been used as a standard model of the business of an insurance company. The constant  is the gross premium rate, the Poisson process .t / counts the number of claims to the company in the time interval Œ0; t , and k is the amount of the kth claim. One of the classical objects of study is the ruin probability of the company that has an initial capital u  0, P .u/ D Pf inf u .t / < 0g:

(7.1.2)

t0

Of a special interest is the asymptotics of the ruin probability P .u/ when the initial capital u tends to infinity. The following constant is a threshold parameter that determines the principal behaviour of the ruin probabilities:  ˛D :  It is known (see, for example, Feller (1966)) that the ruin probability P .u/, considered as a function in u  0, satisfies, in the case where ˛  1, the renewal equation N P .u/ D ˛.1  G.u// C˛ where 1 N G.u/ D 

Z

u

0

Z

u 0

N P .u  s/G.ds/;

u  0;

(7.1.3)

.1  G.s// ds:

N The distribution function that generates this equation is F .u/ D ˛ G.u/. The total variation of the distribution function F .u/ is ˛. Only the case ˛ < 1 is non-trivial, since P .u/  1 is a solution of equation (7.1.3) if ˛ D 1. Note also that in the case where ˛ > 1, the ruin probability is still P .u/  1, but it does not satisfy (7.1.3) in this case.

7.1 Cram´er–Lundberg and diffusion approximations

443

A standard condition under which the Cram´er–Lundberg approximation is valid is the following: C21 : There exists ı > 0 such that: R1 N < 1; (a) 0 e ıs G.ds/ R 1 ıs N (b) ˛ e G.ds/ > 1. 0

Condition C21 guarantees, by Lemma 1.4.3, that the characteristic equation Z 1 N D1 (7.1.4) e s G.ds/ ˛ 0

has a unique positive root . The root  is called the Lundberg exponent. Multiplying equation (7.1.3) by e u transforms it into a proper renewal equation for the function P .u/e u . Applying the renewal theorem (see, for example, Feller 1971) to this equation one obtains the following asymptotic relation known as the Cram´er– Lundberg approximation: P .u/ !  as u ! 1; e u where

(7.1.5)

R1 D

0

N e s .1  G.s// ds R1 : s N 0 se G.ds/

(7.1.6)

7.1.2 Diffusion approximation for ruin probabilities The Cram´er–Lundberg approximation describes the asymptotics of the ruin probabilities for fixed values of ˛ < 1. On the other hand, it does not give a detailed description of the asymptotics for the ruin probability P .u/ for large values of u and values of ˛ that are less than but close to 1. For example, it does not describe the asymptotic behaviour of the ruin probabilities if the initial capital satisfies u ! 1 and, simultaneously, the gross premium rate is  # . Here, the so-called diffusion approximation gives an answer. The asymptotics depends on a balancing relation between the speeds at which  tends to  from above and u tending to infinity. Typical conditions include a balancing condition of the type .1  ˛/u ! % as " ! 0 plus a condition of finiteness of the second moment of the claim distributions. Under these condition, the diffusion approximation yields an asymptotics of the ruin probability in the form P .u/ ! e %=mN as u ! 1; where m N D ˇ=2 and ˇ D EX12 .

(7.1.7)

444

7

Nonlinearly perturbed risk processes

A traditional way to obtain a diffusion type asymptotics is based on approximating the risk processes by a Wiener process with a shift. Weak convergence of the risk processes, normalized in a proper way, to a Wiener process, and the corresponding functional limit theorems for the infinite interval are proved and then used to show that the ruin probabilities for the risk processes can be approximated by the corresponding ruin probabilities for the limit Wiener process with a shift; finally provide explicit formulas to calculate the corresponding ruin probabilities for the limit process are also used. The method described above gave the name for diffusion approximation.

7.2 Diffusion approximation for perturbed risk processes In this section we present asymptotics results concerning the diffusion approximation of the ruin probabilities.

7.2.1 Diffusion approximation of the ruin probabilities for perturbed risk processes ."/

Let u .t /, t  0, be a standard risk process defined for every "  0 in the following way: u."/ .t /

DuC

."/

t

."/ .t/

X

."/

k ;

t  0;

(7.2.1)

kD1 ."/

where (a) u is a nonnegative constant, (b)  ."/ is a positive constant, (c) k , k D 1; 2; : : :, is a sequence of nonnegative i.i.d. random variables with a distribution funcR1 tion G ."/ .u/ that has a finite expectation ."/ D 0 uG ."/ .du/ > 0, (d) ."/ .t /, t  0, is a Poisson process with parameter ."/ > 0, (e) the sequence of the random ."/ variables k , k D 1; 2; : : :, and the process ."/ .t /, t  0 are independent. The object of our interest is the ruin probability P ."/ .u/ D Pf inf u."/ .t / < 0g; t0

u  0:

(7.2.2)

The ruin probability P ."/ .u/ satisfies, for every "  0, the renewal equation (7.1.3) that takes the following form: Z u ."/ ."/ ."/ ."/ N P ."/ .u  s/GN ."/ .ds/; u  0; (7.2.3) P .u/ D ˛ .1  G .u// C ˛ 0

where

."/ ."/ ;  ."/

˛ ."/ D and

1 GN ."/ .u/ D ."/ 

Z 0

u

.1  G ."/ .s// ds:

7.2

Diffusion approximation for perturbed risk processes

445

To ensure that for every "  0, P ."/ .u/ is a solution of (7.2.3), we assume that the following condition is satisfied: H1 :

(a) 0 <  ."/ ; ."/ ; ."/ < 1 for every "  0; (b) ˛ ."/ 2 .0; 1 for every "  0.

The mean value of the distribution function GN ."/ .u/ can be expressed via the first two moments of the claim distribution function G ."/ .u/, m N ."/ D

Z

1 0

uGN ."/ .du/ D

ˇ ."/ ; ˇ ."/ D 2."/

Z 0

1

u2 G ."/ .du/:

(7.2.4)

Let us introduce the following condition: D28 : lim0t!0 lim"!0 G ."/ .t / < 1. Note that, according to the condition E1 , .0/ > 0, we have that G .0/ .0/ < 1 as well. We will also use the following condition: R1 M16 : limT !1 lim0"!0 T u2 G ."/ .du/ D 0. Note that condition M16 reduces to the condition ˇ.0/ < 1 if the claim distributions G ."/ .u/  G .0/ .u/ do not depend on the parameter ". Let f ."/  0 and g ."/ > 0 be functions of "  0. We use the notation f ."/  g ."/ if (a) f ."/ =g ."/ ! 1 as " ! 0 and f ."/  O.g ."/ / if (b) lim"!0 f ."/ =g ."/ > 0, and (c) lim"!0 f ."/ =g ."/ < 1. The following condition is a variant of the balancing condition: B9 : u."/ ! 1 and ˛ ."/ ! 1 in such a way that .1  ˛ ."/ /u."/  O.1/ as " ! 0. The main result of this section is the following theorem. Theorem 7.2.1. Let conditions H1 , D28 , M16 , and B9 hold. Then P ."/ .u."/ / e .1˛ ."/ /u."/ =mN ."/

! 1 as " ! 0:

(7.2.5)

Proof. We prove several lemmas that are weak variants of Theorem 7.2.1. These lemmas have their own value and divide the proof of Theorem 7.2.1 into several logical steps. The first lemma is based on the following conditions: N N D29 : GN ."/ ./ ) G./ as " ! 0, where G.u/ is a non-arithmetic distribution function;

446

7

and M17 : m N ."/ ! m N D

R1 0

Nonlinearly perturbed risk processes

N uG.du/ < 1 as " ! 0.

Remark 7.2.1. As was shown in Remark 1.2.2, condition M17 is equivalent, under condition D29 , to the following relation: Z 1 lim lim uGN ."/ .du/ D 0: (7.2.6) T !1 0"!0 T

Let us also use the following variant of the balancing condition: B10 : u."/ ! 1 and ˛ ."/ ! 1 in such a way that .1  ˛ ."/ /u."/ ! % 2 Œ0; 1 as " ! 0. Lemma 7.2.1. Let conditions H1 , D29 , M17 , and B10 hold. Then P ."/ .u."/ / ! e %=mN as " ! 0:

(7.2.7)

Proof. We are going to apply Theorem 1.3.1 to the renewal equation (7.2.3). Conditions B10 and D29 imply that condition D6 holds for the distribution functions N F ."/ .u/ D ˛ ."/ GN ."/ .u/ with the limit distribution function F .0/ .u/ D G.u/. Conditions B10 and M17 imply that condition M6 holds for the distribution functions F ."/ .u/ with m N being a limit value of the expectation. In this case, the free term in equation (7.2.3) is q ."/ .u/ D ˛ ."/ .1  GN ."/ .u//. Monotonicity of this function and conditions D29 and B10 imply in an obvious way that condition F2 .a/ holds. The set of points of continuity of the limit function N can serve as a set of local uniform convergence in condition F2 .a/. q0 .u/ D 1  G.u/ Condition F2 .b/ holds, since 0  q ."/ .u/  1. Condition F2 .c/ follows from (7.2.6) and the following estimate: X X sup j q ."/ .u/ j D h ˛ ."/ .1  GN ."/ .rh// h rT = h r hu.rC1/h

Z  

rT = h 1

T h Z 1 T h

.1  GN ."/ .s// ds s GN ."/ .ds/:

Condition B10 coincides with B1 . Finally, the limit constant x .0/ .1/ takes the following form: R1 N .1  G.u// du .0/ x .1/ D 0 R 1 D 1: N uG.du/

(7.2.8)

(7.2.9)

0

The proof can be completed by applying Theorem 1.3.1 to the renewal equation (7.2.3).

7.2

Diffusion approximation for perturbed risk processes

447

We can now improve conditions of Lemma 7.2.1 by using the conditions formulated directly in terms of the claim distributions G ."/ .u/. Let us assume that the following condition is satisfied: D30 : G ."/ ./ ) G .0/ ./ as " ! 0. Note that condition H1 implies that the distribution function G .0/ .u/ is not concentrated in zero. Lemma 7.2.2. Let conditions H1 , D30 , M16 and B10 hold. Then .0/

P ."/ .u."/ / ! e %=mN

as " ! 0;

(7.2.10)

where m N .0/ D ˇ .0/ =2.0/ . Proof. Conditions D30 and M16 obviously imply that ."/ ! .0/ as " ! 0:

(7.2.11)

Condition H1 implies that .0/ > 0 and, therefore, G .0/ .0/ < 1. This relation and condition D30 imply that condition D28 holds. Condition D30 and relation (7.2.11) imply that GN ."/ ./ ) GN .0/ ./ as " ! 0:

(7.2.12)

Since the distribution function G .0/ .u/ is not concentrated in zero, the distribution function GN .0/ .u/ has a non-zero absolutely continuous component and, hence, it is non-arithmetic. Relation (7.2.12) and the remark above imply that condition D29 holds for the limit N distribution function G.u/ D GN .0/ .u/. With D30 , condition M16 is equivalent to the following relation: ˇ."/ ! ˇ .0/ D

Z

1 0

u2 G .0/ .du/ < 1 as " ! 0:

(7.2.13)

Relations (7.2.11) and (7.2.13) imply that m."/ ! m.0/ D ˇ .0/ =2.0/ as " ! 0:

(7.2.14)

N D Relation (7.2.14) implies that condition M17 holds with the limit constant m m.0/ . The proof can now be completed by applying Lemma 7.2.1.

448

7

Nonlinearly perturbed risk processes

We can now weaken the conditions of Lemma 7.2.2 by omitting the conditions of weak convergence of the corresponding distributions used in this lemma. Let us now assume the following condition: N ."/ ! m N as " ! 0. M18 : m N 2 .0; 1/ in condition M18 . Note that conditions D28 and M16 imply that m Lemma 7.2.3. Let conditions H1 , D28 , M16 , B10 , and M18 hold. Then P ."/ .u."/ / ! e %=mN as " ! 0:

(7.2.15)

Proof. Let "n > 0 be an arbitrary subsequence such that "n ! 0 as n ! 1. One can always select a subsequence "0k D "nk ! 0 as k ! 1 from the subsequence "n such that 0

G ."k / ./ ) G./ as k ! 1;

(7.2.16)

where G.u/ is a distribution function, possibly improper or concentrated in zero. Conditions D28 and M16 imply, however, that the limit distribution function G.u/ is proper and not concentrated in zero. Moreover, relation (7.2.16) and condition M16 imply that Z 1 ."0k / !D uG.du/ as k ! 1; (7.2.17)  0

and ˇ

."0k /

Z !ˇD

0

1

u2 G.du/ as k ! 1:

(7.2.18)

Relations (7.2.17) and (7.2.18), formula (7.2.4), and condition M18 imply that 0

m N ."k / ./ ! m N D ˇ=2 as k ! 1:

(7.2.19)

Note that the limit distribution G.u/ in (7.2.16), as well as the constants  and ˇ in (7.2.17) and (7.2.18), could depend on the choice of the subsequences "n and "0k , but the constant ˇ=2 must be equal to m, N due to condition M18 and, therefore, it does not depend on the choice of the subsequences "n and "0k . ."0 /

Now we can apply Lemma 7.2.2 to the risk processes u k .t /, t  0, and get the following relation: 0

0

P ."k / .u."k / / ! e %=mN as k ! 1:

(7.2.20)

Since the choice of the initial subsequence "n was arbitrary, relation (7.2.20) implies that P ."/ .u."/ / ! e %=mN as " ! 0: The proof is complete.

(7.2.21)

7.2

Diffusion approximation for perturbed risk processes

449

Now we can finish the proof of Theorem 7.2.1. The last step is to weaken conditions B10 and M18 . So, we assume that conditions H1 , D28 , M16 , and B9 hold. Let "n > 0 be an arbitrary subsequence such that "n ! 0 as n ! 1. One can always select a subsequence "0k D "nk ! 0 as k ! 1 from the subsequence "n such that 0

m N ."k / ! m N as " ! 0;

(7.2.22)

where m N is a constant from the interval Œ0; 1. N 2 Conditions D28 and M16 imply, however, that the limit constant satisfies m .0; 1/. Now one can select a subsequence "00r D "0kr ! 0 as r ! 1 from the subsequence 0 "k such that 00

00

.1  ˛ ."r / /u."r / ! % as " ! 0;

(7.2.23)

where % is a constant from the interval Œ0; 1. Condition B9 implies, however, that % 2 .0; 1/. ."00 / Now we can apply Lemma 7.2.3 to the risk processes u r .t /, t  0, and get the following relation: 00

00

P ."r / .u."r / / ! e %=mN as r ! 1:

(7.2.24)

Due to (7.2.22) and (7.2.23), relation (7.2.24) can be rewritten in an equivalent form, 00

00

00

00

P ."r / .u."r / / P ."r / .u."r / /  ! 1 as r ! 1: ."00 / ."00 / ."00 / e %=mN e .1˛ r /u r =mN r

(7.2.25)

Since the initial subsequence "n was chosen in an arbitrary way, relation (7.2.25) is equivalent to the relation P ."/ .u."/ / e .1˛

."/ /u."/ =m N ."/

! 1 as " ! 0:

(7.2.26)

The proof is complete. It should be noted that the conditions of Theorem 7.2.1 do not require convergence of the parameters  ."/ and ."/ , and the weak convergence of the claim distributions G ."/ ./. In the case where the parameters  ."/ and ."/ converge to some limit values  .0/ and .0/ , respectively, and condition D30 holds, one can consider the process ."/ .0/ u .t /, t  0, as a perturbation of the limit risk process u .t /, t  0, with the gross premium rate  .0/ , the intensity of the claim flow .0/ , and the claim distribution function G .0/ .u/.

450

7

Nonlinearly perturbed risk processes

7.2.2 Asymptotics for increments of ruin probabilities Let us now study the asymptotics of increments of the ruin probabilities P ."/ .u" Ct / P ."/ .u" C t C h/, where t 2 R1 , h > 0. As usual we assume that the ruin probability is P" .u/ D 0 for u < 0. The following theorem supplements Theorem 7.2.1. Theorem 7.2.2. Let conditions H1 , D28 , M16 , and B9 hold. Then, for any t 2 R1 , h > 0, P ."/ .u."/ C t /  P ."/ .u."/ C t C h/ .1  ˛ ."/ /e .1˛ ."/ /u."/ =mN ."/ mN h."/

! 1 as " ! 0:

(7.2.27)

Proof. Introduce the renewal function generated by the distribution function ."/ ˛ ."/ GN n .u/, 1 X ."/ U .u/ D .˛ ."/ /n GN n."/ .u/; u  0; nD0 ."/ GN n .u/ is ."/

where the n-fold convolution of the distribution function GN ."/ .u/. Let also U .A/ be a renewal measure on the Borel -algebra of Œ0; 1/, which is determined by its values on the intervals U ."/ ..a; b/ D U ."/ .b/  U ."/ .a/, 0  a  b < 1. Note that this measure has the atom 1 in the point 0, since the 0-fold ."/ convolution GN 0 .s/, by the definition, is a distribution function that has a unit jump in the point 0. The renewal equation (7.2.3) has a unique solution given that is given by the same formula as for the proper renewal equation, Z u P ."/ .u/ D ˛ ."/ .1  GN ."/ .u  s//U ."/ .ds/; u  0: (7.2.28) 0

Standard calculations permit to transform this formula into the following form: P

."/

.u/ D

1 Z X nD0 0

D ˛ ."/ 

u

˛ ."/ .1  GN ."/ .u  s//.˛ ."/ /n GN n."/ .ds/

1 X

.1  ˛ ."/ /.˛ ."/ /n GN n."/ .u/ D 1  .1  ˛ ."/ /U ."/ .u/:

nD1

This formula can be rewritten in the form, 1  P ."/ .u/ D .1  ˛ ."/ /U ."/ .u/:

(7.2.29)

Formula (7.2.29) is the famous Khintchine–Pollaczek formula. It permits to reduce studies of the asymptotics for the increments of ruin probabilities to studies of an asymptotics for the increments of the renewal function.

7.2

Diffusion approximation for perturbed risk processes

451

Let us consider the renewal equation with the forcing function q ."/ .u/ DŒth;t .u/, u  0, Z u ."/ ."/ x .u/ D Œth;t .u/ C ˛ x ."/ .u  s/GN ."/ .ds/; u  0: (7.2.30) 0

Taking into account the indicator form of the forcing function and formula (7.2.29), we can represent the solution of this equation in the following form: Z u ."/ x .u/ D Œth;t .u  s//U ."/ .ds/ D U ."/ .u  t C h/  U ."/ .u  t / 0

D .P ."/ .u  t /  P ."/ .u  t C h//=.1  ˛ ."/ /;

u  0:

(7.2.31)

Theorem 7.2.2 is an analogue of Theorem 7.2.1, in which the functions x ."/ .u/ replace the ruin probabilities P ."/ .u/. The proof of Theorem 7.2.2 repeats the main steps of the proof of Theorem 7.2.1. The renewal equations (7.2.30) and (7.2.3) have, respectively, solutions x ."/ .u/ and P ."/ .u/. These equations have the same generating distribution function ˛ ."/ GN ."/ .s/ and forcing functions. These are, respectively, the functions R u differ only in the ."/  .u/ and ˛ .1  GN ."/ .u//. This permits to formulate and prove lemmas 0 Œth;t similar to Lemmas 7.2.1, 7.2.2, and 7.2.3. For the lemmas formulated below, we replace U ."/ .u/ with .1P ."/ .u//=.1˛ ."/ / in the formula defining that defines the function x ."/ .u/. Also, we use the argument u."/ C 2t instead of u."/ . This does not change the corresponding balancing conditions B9 and B10 . Lemma 7.2.4. Let conditions H1 , D29 , M17 , and B10 hold. Then, for any t 2 R1 , h > 0, P ."/ .u."/ C t /  P ."/ .u."/ C t C h/ h ! e %=mN as " ! 0: ."/ m N .1  ˛ /

(7.2.32)

The proof is similar to the proof of Lemma 7.2.1. The difference is only in that the forcing function ˛" .1 GN ."/ .u// has to be replaced with a simpler function Œth;t .u/. This function obviously satisfies condition F2 . Lemma 7.2.5. Let conditions H1 , D30 , M16 and B10 hold. Then, for any t 2 R1 , h > 0, P ."/ .u."/ C t /  P ."/ .u."/ C t C h/ h ! e %=mN 0 as " ! 0: ."/ m N .1  ˛ / 0

(7.2.33)

Lemma 7.2.6. Let conditions H1 , D28 , M16 , B10 hold. Then, for any t 2 R1 , h > 0, h P ."/ .u."/ C t /  P ."/ .u."/ C t C h/ as " ! 0: ! e %=mN ."/ m N .1  ˛ /

(7.2.34)

452

7

Nonlinearly perturbed risk processes

The proofs of Lemmas 7.2.5 and 7.2.6 are absolutely analogous to the proofs of Lemmas 7.2.2 and 7.2.3. Using Lemma 7.2.6 we can also complete the proof of Theorem 7.2.2 by repeating the selection arguments that were used in the proof of Theorem 7.2.1.

7.2.3 Asymptotics for densities of non-ruin probabilities We will also give asymptotics for densities of non-ruin probabilities. Note that the non-ruin probability 1  P ."/ .u/ is a distribution function on Œ0; 1/ with the jump 1  ˛ ."/ in zero. The equation (7.2.3) can be rewritten as 1  P ."/ .u/ D 1  ˛ ."/ C˛

."/

Z

u 0

.1  P ."/ .u  s//

1  G ."/ .s/ ds; ."/

u  0:

(7.2.35)

If 1  G ."/ .s/ were a continuous function we could obtain, by “formally” differentiating (7.2.35), that 1  P ."/ .u/ has a derivative p ."/ .u/ that satisfies the following equation: p ."/ .u/ D .1  ˛ ."/ /˛ ."/ C ˛ ."/

Z 0

u

1  G ."/ .u/ ."/ p ."/ .u  s/

1  G ."/ .s/ ds; ."/

u  0:

(7.2.36)

The renewal equation (7.2.36) has a unique solution even if the function 1 G ."/ .s/ is to be continuous. One can easily check that the function 1  ˛ ."/ C R unot assumed ."/ ."/ 0 p .s/ ds, u  0, satisfies equation (7.2.35) and, hence, p .u/ is indeed a density of the distribution function 1  P ."/ .u/ (without the the assumption that 1  G ."/ .s/ is continuous), and Z u ."/ ."/ p ."/ .s/ ds; u  0: (7.2.37) 1  P .u/ D 1  ˛ C 0

It follows from (7.2.36) that p" .u/ is a c`adl`ag function. It is given by the formula Z u 1  G ."/ .u  s/ ."/ ."/ ."/ ."/ p .u/ D .1  ˛ /˛ U .ds/; u  0: (7.2.38) ."/ 0 Formula (7.2.38) implies that p ."/ .u/ is a non-negative function. As follows from Lemma 1.1.1 and Remark 1.1.1, conditions H1 and D28 imply that there exists "0 > 0 such that for any h > 0, sup

sup

""0 0uvuCh

.U ."/ .v/  U ."/ .u// D Kh < 1:

(7.2.39)

7.2

453

Diffusion approximation for perturbed risk processes

Indeed, condition M16 implies that lim ˇ ."/ D B < 1:

(7.2.40)

"!0

Using formula (7.2.38) and relations (7.2.39) and (7.2.40) we get the following estimate for h > 0: p ."/ .u/  lim sup "!0 u0 1  ˛ ."/ "!0 u0

X

lim sup

.1  G ."/ .kh//

kŒu= hC1

 .U ."/ .u  .k  1/h/  U ."/ ..u  kh/ ^ 0// X

 lim sup

"!0 u0

kŒu= hC1

ˇ ."/ BKh Kh  < 1: 2 .kh/ .kh/2

(7.2.41)

."/

.u/ are asymptotically uniformly bounded funcEstimate (7.2.41) means that p 1˛ ."/ tions, as " ! 0. ."/ .u/ The following theorem gives asymptotics for the normalised densities p and 1˛ ."/ supplements Theorem 7.2.2.

Theorem 7.2.3. Let conditions H1 , D28 , M16 , and B9 hold. Then p ."/ .u."/ / .1  ˛ ."/ /e .1˛

."/ /u."/ =m N ."/

1 m N ."/

! 1 as " ! 0:

(7.2.42)

Proof. Equation (7.2.36) can be rewritten in the following form, which is fitted for applying Theorem 1.3.1: ."/ p ."/ .u/ ."/ 1  G .u/ D ˛ 1  ˛ ."/ ."/ Z u p ."/ .u  s/ N ."/ ."/ G .ds/; C˛ 1  ˛ ."/ 0

u  0:

(7.2.43)

."/

.u/ , we can prove an analogue of Theorem 7.2.1 by Now, for the functions p 1˛ ."/ repeating the proof of this theorem. Thus we only formulate analogues of the corresponding lemmas, which are also useful by themselves.

Lemma 7.2.7. Let conditions H1 , D28 , M16 , D30 , and B10 hold. Then p ."/ .u."/ / 1 as " ! 0: ! e %=mN 0 ."/ m N0 .1  ˛ /

(7.2.44)

454

7

Nonlinearly perturbed risk processes

Proof. The proof combines elements of the proofs of Lemmas 7.2.1 and 7.2.2. In the proof of Lemma 7.2.2 it was shown that conditions M16 and D30 imply that condition D29 holds for the distribution functions GN ."/ .u/ with the limit distribution function GN .0/ .u/. In place of the forcing function ˛ ."/ .1  GN ."/ .u//, we have a simpler forcing ."/ .u/ . Condition D30 and relation (7.2.11), obtained in the proof function, ˛ ."/ 1G ."/ of Lemma 7.2.2, imply due to monotonicity of these forcing functions that they sat.0/ .u/ isfy condition F2 .a/. The set of points of continuity of the limit function 1G .0/ can serve as a set of convergence in condition F2 .a/. Condition F2 .b/ holds, since ."/ .u/ 0  ˛ ."/ 1G  1."/ . Condition F2 .c/ follows from the following estimate: ."/ h

X

˛ ."/

sup

r T = h r hu.rC1/h

."/ X 1  G ."/ .u/ ."/ .1  G .rh// D h ˛ ."/ ."/ rT = h Z 1 1  ."/ .1  G ."/ .s// ds  T h

 1  GN ."/ .T  h/ 

m N ."/ : T h

Condition B10 coincides with B1 . Finally, the corresponding renewal limit takes the following form: Z 1 1  G .0/ .u/ 1 1 .0/ x .1/ D .0/ du D .0/ : .0/ m N  m N 0

(7.2.45)

(7.2.46)

The proof can be completed by applying Theorem 1.3.1 to equation (7.2.36). Lemma 7.2.8. Let conditions H1 , D28 , M16 , B10 , and M17 hold. Then p ."/ .u."/ / 1 as " ! 0: ! e %=mN ."/ m N .1  ˛ /

(7.2.47)

The proof of Lemma 7.2.8 is absolutely similar to the proof of Lemma 7.2.3. Using Lemma 7.2.8 we can also complete the proof of Theorem 7.2.3 by repeating the selection arguments used in the proof of Theorem 7.2.1.

7.3 Asymptotic expansions in Cram´er–Lundberg approximation for perturbed risk processes In this section we give approximation for perturbed risk processes which generalises the classical Cram´er–Lundberg and diffusion approximations and exponential asymptotic expansions for ruin probabilities for this approximation.

455

7.3 Asymptotic expansions in Cram´er–Lundberg approximation

7.3.1 Generalised Cram´er–Lundberg and diffusion approximations for perturbed risk processes Let us assume that conditions H1 and D30 hold. Let us also introduce conditions: D31 :  ."/ !  .0/ as " ! 0, and D32 : ."/ ! .0/ as " ! 0. Consider the following moment generating function: Z 1 Z ."/ s ."/ ."/ e .1  G .s// ds D  ' ./ D 0

1 0

e s GN ."/ .ds/:

The following condition is an analogue of the Cram´er type condition C2 : C22 : There exists ı > 0 such that: (a) lim0"!0 ' ."/ .ı/ < 1; R1 .0/ (b) c .0/ ' .0/ .ı/ D ˛ .0/ 0 e ıs GN .0/ .ds/ 2 .1; 1/. Condition C22 obviously implies condition M16 . As was pointed out in the proof of Lemma 7.2.2, conditions H1 , D30 and M16 imply that (a) ."/ ! .0/ as " ! 0; (b) GN ."/ ./ ) GN .0/ ./ as " ! 0; and that (c) GN .0/ .s/ is a non-arithmetic distribution function. Relation (b) and conditions H1 and C22 imply that (d) ' ."/ ./ ! ' .0/ ./ as " ! 0, for  < ı. Statement (a) and conditions D31 and D32 obviously imply that (e) ˛ ."/ ! ˛ .0/ as " ! 0. Let us consider the following characteristic equation: ."/ c ."/

Z

1 0

e s .1  G ."/ .s// ds D 1:

(7.3.1)

Relations (b) and (e), and condition C22, with a use of Lemma 1.4.3, imply that (f) there exists "1 > 0 such that the characteristic equation (7.3.1) for every "  "1 has a unique non-negative root ."/ , and (g) ."/ ! .0/ as " ! 0. For n D 0; 1; : : :, let us also introduce the mixed power-exponential moment generating functions Z 1 Z 1 ' ."/ Œ; n D s n e s .1  G ."/ .s// ds D ."/ s n e s GN ."/ .ds/: 0

By the definition, ' ."/ Œ; 0 D ' ."/ ./.

0

456

7

Nonlinearly perturbed risk processes

Let us choose an arbitrary .0/ < ˇ < ı. Condition C22 (a) implies that (h) there exists "2 D "2 .ˇ/ > 0 such that, for "  "2 and n D 0; 1; : : :, Z 1 ."/ ' Œˇ; n  cn e ıs .1  G ."/ .s// ds < 1; (7.3.2) 0

where cn D cn .ı; ˇ/ D sups0 s n e .ıˇ /s < 1. Note also that ' ."/ Œ; n, for   ˇ, is the derivative of order n of the function ."/ ' ./. Denote R 1 ."/ s e .1  GN ."/ .s// ds ."/ ."/  . / D 0 R 1 ."/ s G N ."/ .ds/ 0 se R 1 ."/ s R 1 e . .1  G ."/ .u// du/ ds : (7.3.3) D 0 R 1 s."/  s .1  G ."/ .s/ ds 0 se Relation (g) implies that (i) there exists "3 D "3 .ˇ/ > 0 such that ."/ < ˇ. Define "0 D min."1 ; "2 ; "3 /:

(7.3.4)

R 1 ."/ Relations (b) and (g), and conditions H1 and C22 imply that (j) 0 e  s .1  R1 ."/ GN ."/ .s// ds < 1 and (k) 0 se  s GN ."/ .ds/ < 1 for "  "3 . Therefore, the quantity  ."/ .."/ / is well defined for all "  "3 . The following theorem generalises both the classical approximations for ruin probabilities, namely the Cram´er–Lundberg approximation given by the asymptotic relation (7.1.5), and the diffusion approximation given by the asymptotic relation (7.1.7). Theorem 7.3.1. Let conditions H1 , D30 –D32 , and C22 hold. Then for any 0  u."/ ! 1 as " ! 0 the following asymptotic relation holds: P ."/ .u."/ / !  .0/ ..0/ / as " ! 0: expf."/ u."/ g

(7.3.5)

Proof. We can apply Theorem 1.4.2 to the renewal equation (7.2.3). In this case, the forcing function is q ."/ .u/ D ˛ ."/ .1  GN ."/ .u// and the distribution function that generates this equation is F ."/ .u/ D ˛ ."/ GN ."/ .u/. Conditions H1 and D30 –D32 imply, as was mentioned in Subsection 7.3.1, that (b) GN ."/ ./ ) GN .0/ ./ as " ! 0, (c) the limit distribution function GN .0/ .u/ is nonarithmetic, and (e) ˛ ."/ ! ˛ .0/ as " ! 0. Relations (b), (c), and (e) imply that condition D7 holds. Note that, according condition H1 , the parameter f ."/ D 1  ˛ ."/ 2 Œ0; 1/.

7.3 Asymptotic expansions in Cram´er–Lundberg approximation

457

Let first consider the case, where .0/ > 0. Note that (l) .0/ > 0 if and only if ˛ .0/ < 1, Condition C2 takes in this case the form of condition C22 . Monotonicity of the function q ."/ .u/ D ˛ ."/ .1  GN ."/ .u// and relations (b), (e), and (g) ."/ ! .0/ as " ! 0 imply in an obvious way that condition F6 .a/ holds. The set of points of continuity of the limit function q .0/ .u/ D ˛ .0/ .1  GN .0/ .u// can serve as a set of local uniform convergence in condition F6 .a/. Condition F6 .b/ holds, since 0  q ."/ .u/  1. As was mention above, conditions H1 , D30 –D32 , and C22 imply that Z 1 e ˇs .1  GN ."/ .s// ds < 1: (7.3.6) lim "!0 0

Using relation (7.3.6) we get for any .0/ < ˇ  ı and   ˇ  .0/ that X .0/ lim h sup e . C /u j q ."/ .u/ j "!0

r T = h r hu.rC1/h

 lim h "!0

e

ˇh

X

e .

.0/ C /.r C1/h

˛ ."/ .1  GN ."/ .rh//

r T = h

Z

lim

1

"!0 T h

e ˇs .1  GN ."/ .s// ds ! 0 as T ! 1:

(7.3.7)

Relation (7.3.7) implies that condition F6 .c/ also holds. Let now consider the case, where .0/ D 0. Note that (m) .0/ D 0 if and only if .0/ ˛ D 1. Condition C1 takes the form of condition C22 (a) if .0/ D 0. Condition F5 is verified in the same way as it was done above for condition F6 . Finally, the renewal limit xQ .0/ .1/ takes the form of the quantity  .0/ ..0/ / in formula (7.3.3). The proof can be completed by applying Theorem 1.4.2 to the renewal equation (7.2.3). The statements (i) or (ii) of this theorem should be used in the cases .0/ > 0 and .0/ D 0, respectively. As was mentioned in the proof, (l) .0/ > 0 if and only if ˛ .0/ < 1, while (m) D 0 if and only if ˛ .0/ D 1. The case (l) .0/ > 0 corresponds to a model of the Cram´er–Lundberg approximation, while the case (m) .0/ D 0 corresponds to a model of the diffusion approximation. In the case (m) .0/ D 0, Theorem 7.3.1 impose stronger condition on the claim distributions then Lemma 7.2.2, which is, in fact, a direct analogue of the classical diffusion approximation in the case of perturbed risk processes. Indeed, conditions C22 and M16 are used in Theorem 7.3.1 and Lemma 7.2.2, respectively. However, no balancing condition that would restrict the rate of growth of the time parameter u."/ to 1 is assumed in Theorem 7.3.1, while the balancing condition B10 is assumed in Lemma 7.2.2. .0/

458

7

Nonlinearly perturbed risk processes

Note also that, in the case (m) .0/ D 0, conditions H1 , D30 and C22 imply, by Lemma 1.4.2 and formula (7.2.4), the following asymptotic relation ."/  .1  ˛ ."/ / 

2."/ as " ! 0: ˇ ."/

(7.3.8)

7.3.2 Perturbation conditions for risk processes Let us formulate and clarify a connection between different forms of perturbation conditions for the risk processes. We assume that condition H1 holds. The perturbation conditions for the parameters  ."/ and ."/ are formulated in the following way: .k/ P39 :  ."/ D  .0/ C d1 " C    C dk "k C o."k /, where jdl j < 1, l D 1; : : : ; k,

and .k/

P40 : ."/ D .0/ C e1 " C    C ek "k C o."k /, where jel j < 1, l D 1; : : : ; k. It is also convenient to define d0 D  .0/ and e0 D .0/ . Note that, according to condition E1 , we have (a)  .0/ ; .0/ > 0. .k/ .k/ Also, conditions P39 and P40 imply that conditions D31 and D32 hold. Let us also introduce the following perturbation condition for  < ˇ: .;k/

: ' ."/ Œ; n D ' .0/ Œ; n C vŒ; 1; n" C    C vŒ; k  n; n"kn C o."kn /, where j vŒ; r; n j < 1 for r D 1; : : : ; k  n, n D 0; : : : ; k. It is also convenient to set vŒ; 0; n D ' .0/ Œ; n, n D 0; : : : ; k. The distribution function, which generates the renewal equation (7.2.3), is F ."/ .u/ D ˛ ."/ GN ."/ .u/. In order to apply theorems given in Chapter 1, we should also obtain asymptotic expansions for the corresponding mixed power-exponential generating functions, Z 1 ."/ 'Q ."/ Œ; n D ˛ ."/ s n e s GN ."/ .ds/ D ."/ ' ."/ Œ; n: (7.3.9)  0 P41

.k/

.k/

.;k/

Lemma 7.3.1. Let conditions H1 , C22 , P39 , P40 hold and P41   ˇ. Then the following asymptotic expansion takes place:

be satisfied for some

'Q ."/ Œ; n D 'Q .0/ Œ; n C bŒ; 1; n" C    C bŒ; k  n; n"kn C o."kn /; (7.3.10) where the coefficients bŒ; r; n are given by the recurrence formulas bŒ; 0; 0 D .0/ ' .0/ Œ;n D e0 vŒ;0;n and, for r D 0; : : : ; k  n and given n  k and sequend0  .0/ tially for, n D 0; : : : ; k, bŒ; r; n D

r X mD0

hm vŒ; r  m; n;

(7.3.11)

459

7.3 Asymptotic expansions in Cram´er–Lundberg approximation

where   m X hm D d01 em  dq hmq :

(7.3.12)

qD1

Proof. The proof follows from formula (7.3.9) and statements (v) and (iii) of Lemma ."/ ."/ 8.1.1, which should be applied first to the quotient ."/ and then to the product ."/  ' ."/ Œ; n. Remark 7.3.1. The first coefficients bŒ; r; n, r D 0; 1; 2 and n D 0; 1; : : :, which are used below with n D 0; 1; 2 for calculating the coefficients a1 and a2 in the corresponding expansions for characteristic roots ."/ , are given by the following formulas: e0 vŒ; 0; n; d0 e1  d1 h0 e0 vŒ; 1; n C vŒ; 0; n bŒ; 1; n D d0 d0 d0 e1  d1 e0 e0 vŒ; 1; n C vŒ; 0; n; D d0 d02 bŒ; 0; n D

bŒ; 2; n D

e0 e1  d1 h0 vŒ; 2; n C vŒ; 1; n d0 d0 e2  d1 h1  d2 h0 vŒ; 2; n: C d0

(7.3.13)

Finally, let us introduce the following balancing condition: .r/

B11 : 0  u."/ ! 1 in such a way that "r u."/ ! %r , where 0  %r < 1.

7.3.3 Exponential expansions for ruin probabilities Now we are in a position to apply theorems of Chapter 1 to the perturbed renewal equation (7.2.3). .k/

.k/

..0/ ;k/

Theorem 7.3.2. Let conditions H1 , D30 , C22 , P39 , P40 , and P41

hold. Then

(i) The root ."/ of equation (7.3.1) has the asymptotic expansion ."/ D .0/ C a1 " C    C ak "k C o."k /;

(7.3.14) .0/

;1;0 where the coefficients an are given by the recurrence formulas a1 D  bŒ bŒ.0/ ;0;1

460

7

Nonlinearly perturbed risk processes

and in general for n D 1; : : : ; k,  n1 X an D  bŒ.0/ ; 0; 11 bŒ.0/ ; n; 0 C bŒ.0/ ; n  q; 1aq qD1

C

X

n X

bŒ

.0/

; n  q; m 

2mn qDm

X

q1 Y

 ; (7.3.15)

n app =np Š

n1 ;:::;nq1 2Dm;q pD1

where the coefficients bŒ.0/ ; r; n are given by (7.3.11) and Dm;q , for every 2  m  q < 1, is the set of all nonnegative, integer solutions of the system n1 C    C nq1 D m;

n1 C    C .q  1/nq1 D q:

(7.3.16)

(ii) If bŒ.0/ ; l; 0 D 0, l D 1; : : : ; r, for some 1  r  k, then a1 ; : : : ; ar D 0. If bŒ.0/ ; l; 0 D 0, l D 1; : : : ; r  1, but bŒ.0/ ; r; 0 < 0 for some 1  r  k, then a1 ; : : : ; ar 1 D 0 but ar > 0. .r/

(iii) If, additionally, condition B11 is satisfied for some 1  r  k, then the following asymptotic relation holds: P ."/ .u."/ / expf..0/ C a1 " C    C ar1 "r1 /u."/ g ! e %r ar  .0/ ..0/ / as " ! 0:

(7.3.17)

Proof. We can apply Theorem 2.2.1 to the renewal equation (7.2.3). In this case, the forcing function is q ."/ .u/ D ˛ ."/ .1  GN ."/ .u// and the distribution function that generates this equation is F ."/ .u/ D ˛ ."/ GN ."/ .u/. .k/ As was mentioned above, conditions H1 , D30 , and the perturbation conditions P39 , .k/

..0/ ;k/

imply that conditions D31 and D32 hold. Therefore, all conditions of P40 , and P41 Theorem 7.3.1 hold. As was shown in the proof of this theorem, condition D7 holds. This condition coincides in this case with condition D11 . Also, it was show in the proof of Theorem 7.3.1 that condition F6 holds. Condition C2 takes in this case the form of condition C22 . .k/ .k/ .k/ ..0/ ;k/ that follows from Condition P3 is implied by conditions P39 , P40 , and P41 Lemma 7.3.1. Formulas for the coefficients in the corresponding asymptotic expansion are given by formula (7.3.11). .r/ .r/ The balancing condition B3 takes in this case the form of condition B11 . Finally, the renewal limit x .0/ .1/ takes the form of the quantity  .0/ ..0/ / in formula (7.3.3). The proof of the theorem can be completed by applying Theorem 2.2.1 to the renewal equation (7.2.3).

461

7.3 Asymptotic expansions in Cram´er–Lundberg approximation

Remark 7.3.2. The coefficients a1 and a2 can be calculated with the use of formulas (2.2.6) given in Subsection 2.2.1, where the coefficients br;n should be replaced with the coefficients bŒ.0/ ; r; n calculated with the use of formulas (7.3.13) given in Subsection 7.3.1, i.e., bŒ.0/ ; 1; 0 ; bŒ.0/ ; 0; 1 1 1 a2 D  .bŒ.0/ ; 2; 0 C bŒ.0/ ; 1; 1a1 C bŒ.0/ ; 0; 2a12 / .0/ 2 bŒ ; 0; 1 a1 D 

bŒ.0/ ; 1; 1bŒ.0/ ; 1; 0 bŒ.0/ ; 0; 2bŒ.0/ ; 1; 02 bŒ.0/ ; 2; 0  ; C bŒ.0/ ; 0; 1 bŒ.0/ ; 0; 12 2bŒ.0/ ; 0; 13 1 a3 D  .bŒ.0/ ; 3; 0 C bŒ.0/ ; 2; 1a1 C bŒ.0/ ; 1; 1a2 .0/ bŒ ; 0; 1 1 1 C bŒ.0/ ; 1; 2a12 C bŒ.0/ ; 0; 2a1 a2 C bŒ.0/ ; 0; 3a13 /: (7.3.18) 2 6 D

Let us also formulate an analogue of Theorem 2.2.2 with regard to the renewal equation (7.2.3). Introduce the integer parameters 1  h  k0 and kr  1; r D 1; : : :, and define the vector parameter kN D .h; k0 ; : : : ; knQ /, where nQ D max.n W nh  k0 /. Consider the following perturbation condition: N

.;k/ : (a) ' ."/ Œ; 0 D ' .0/ Œ; 0 C vŒ; h; 0"h C    C vŒ; k0 ; 0"k0 C o."k0 /; P42

where jvŒ; r; 0j < 1; r D 1; : : : ; k0 ; (b) ' ."/ Œ; n D ' .0/ Œ; n C vŒ; 1; n" C    C vŒ; kn ; n"kn C o."kn /, where jvŒ; r; nj < 1; r D 1; : : : ; kn ; n D 1; : : : ; n. Q As above, it will also be convenient to set vŒ; 0; n D ' .0/ Œ; n, n D 0; : : : ; k. .;k/ .k/ is more general than condition P41 , and the former reduces to the Condition P42 latter if h D 1, k0 D k (in this case, nQ D k) and kn D k  n, r D 1; : : : ; n. Q The following theorem generalises Theorems 7.3.2. The proof is the same as the one given for Theorems 7.3.2 and is based on applying Theorem 2.2.2 to the renewal equation (7.2.3). .k/

.k/

..0/ ;kQ /

Theorem 7.3.3. Let conditions H1 , D30 , C22 , P39 , P40 , and P42

hold. Then

(i) For all " small enough there exists a unique nonnegative root ."/ of equation (7.3.1) having the following asymptotic expansion: ."/ D .0/ C ah "h C    C ak "k C o."k /;

(7.3.19)

462

7

Nonlinearly perturbed risk processes

where k D min.k0 ; k1 C h; : : : ; knQ C hn/ Q and the coefficients an ; n D h; : : : ; k, are given by the recurrence formulas ah D bŒ.0/ ; h; 0=bŒ.0/ ; 0; 1 and, for n D h; : : : ; k, an D  bŒ

.0/

; 0; 1

1

 bŒ

.0/

; n; 0 C

n1 X

bŒ.0/ ; n  i; 1ai

iDh

C

Œn= Xh

n X

bŒ

.0/

; n  j; r 

r D2 j Dr h

X

jY 1



aini =ni Š

; (7.3.20)

nh ;:::;nj 1 2Dh;r;j iDh

where the coefficients bŒ.0/ ; r; n are given by formula (7.3.11), and Dh;r;j , for every h  1 and 2  r  j < 1, is the set of all nonnegative, integer solutions of the system nh C    C nj 1 D r; hnh C    C .j  1/nj 1 D j :

(7.3.21)

(ii) If bŒ.0/ ; l; 0 D 0; l D h; : : : ; r for some h  r  k, then a1 ; : : : ; ar D 0. If bŒ.0/ ; l; 0 D 0; l D h; : : : ; r  1, but bŒ.0/ ; r; 0 < 0 for some h  r  k, then ah ; : : : ; ar 1 D 0 but ar > 0. .r/

(iii) If, additionally, conditions B11 hold for some h  r  k, then P ."/ .u."/ / expf..0/ C ah " C    C ar1 "r1 /u."/ g ! e %r ar  .0/ ..0/ / as " ! 0:

(7.3.22)

7.3.4 Modifications of perturbation conditions In this subsection, we consider modifications of the perturbation and other conditions introduced in Subsection 7.3.2. .k/ .k/ Let us first consider conditions P39 and P40 . One of possible interpretations for these conditions is based on an assumption that the intensity of the claim flow,  D ./, is a function of the gross premium rate. Let us assume that (a) ./ has the continuous derivative of an order k, (b) 0 < .0C/  .0/ 1, (c) lim!1 ./ < ˛ .0/  1. Under these assumptions, there exists the  .0/ greatest a largest root of the equation . .0/ /.0/ D ˛ .0/ :  .0/

(7.3.23)

The balance relation connected to the diffusion approximation suggests that the difference    .0/ can be regarded as the parameter ". Note that, due to condition H1 , we need to consider only the values    .0/ .

463

7.3 Asymptotic expansions in Cram´er–Lundberg approximation .k/

The expansion in condition P39 takes the following form:  ."/ D  .0/ C ":

(7.3.24)

.k/

The expansion in condition P40 for the function ."/ D . ."/ / becomes . ."/ / D . .0/ / C 1 . .0/ /" C    C

1 k . .0/ /"k C o."k /; kŠ

(7.3.25)

where r ./ is the derivative of order r of the function ./ for r D 1; : : : ; k. Now, let us consider the perturbation condition P. 42 Introduce the moment generating functions ."/

Z ./ D

1

.0/ ;k Q/

.

e s G ."/ .ds/

0

and the mixed power-exponential moment generating functions for n D 0; 1; : : :, ."/

Z Œ; n D

1 0

s n e s G ."/ .ds/:

By the definition, ."/ Œ; 0 D ."/ ./. The following relation links the moment generating functions ' ."/ ./ and '

."/

Z ./ D

1 0

( D

.

s

e .1  G ."/ Œ; 0

."/

Z .s// ds D E

 1/=

."/ Œ0; 1

."/ ./,

."/

1 0

e s ds

for  ¤ 0, for  D 0.

(7.3.26)

Relation (7.3.26) implies that the asymptotic inequality in condition C22 (a) is equivalent to the following relation: lim

."/

0"!0

.ı/ < 1:

(7.3.27)

Relations (7.3.26) and (7.3.27) imply that, for "  "2 and n D 0; 1; : : :, ."/

Note also that   ˇ.

Z Œˇ; n  cn

."/

1 0

e ıs G ."/ .ds/ D

."/

.ˇ/ < 1:

Œ; n is the derivative of order n of the function

(7.3.28) ."/

./ for

464

7

Nonlinearly perturbed risk processes

The following relation connects the derivatives of the functions ' ."/ Œ; n and ."/ Œ; n for n D 1; 2; : : :, "  " , and   ˇ: 2 ( . ."/ Œ; n  n' ."/ Œ; n  1/= for  ¤ 0;   ˇ, (7.3.29) ' ."/ Œ; n D ."/ Œ0; n C 1=.n C 1/ for  D 0. Relation (7.3.29) can be obtained similarly to relation (5.5.43). Relation (a) Œ; 0 D ' ."/ Œ; 0 C 1,   ˇ, follows from formula (7.3.26). Differentiating (a) yields the following relation: (b) ."/ Œ; n D ' ."/ Œ; n C n' ."/ Œ; n  1, n D 1; 2; : : :,   ˇ. In particular, (b) yields the following relation for  D 0: (c) ."/ Œ0; n D n' ."/ Œ0; n  1, n D 1; 2; : : :,   ˇ. Relations (a)–(c) imply relation (7.3.29). .;k/ Let us formulate a perturbation condition that would imply condition P41 , ."/

.;kC1/

P43

:

."/

Œ; n D .0/ Œ; n C uŒ; 1; n" C    C uŒ; k C 1  n; n"kC1n C o."kC1n /, where juŒ; r; nj < 1, r D 1; : : : ; k C1n, n D 0; : : : ; k C1.

It also is convenient to define uŒ; 0; n D .0/ Œ; n, n D 0; : : : ; k. The following lemma is an obvious corollary of relations (7.3.26) and (7.3.29). .;kC1/

Lemma 7.3.2. Let condition H1 , C22 hold, and condition P43 be satisfied for .;k/ some   ˇ. Then condition P41 holds for   ˇ, and the asymptotic expansions in this condition take the following form for n D 0; : : : ; k: ' ."/ Œ; n D ' .0/ Œ; n C vŒ; 1; n" C    C vŒ; k  n; n"kn C o."kn /; (7.3.30) where vŒ; r; n, r D 0; : : : ; k  n, n D 0; : : : ; k, for r D 0; : : : ; k, n D 0, are given by the formulas ( .uŒ; r; 0  ı.r; 0//= if  ¤ 0, (7.3.31) vŒ; r; 0 D uŒ0; r; 1 if  D 0, and, for r D 0; : : : ; k  n, n D 1; : : : ; k, ( .uŒ; r; n  nvŒ; r; n  1/= vŒ; r; n D uŒ0; r; n C 1=.n C 1/

if  ¤ 0, if  D 0.

.;kC1/

(7.3.32) .;k/

Remark 7.3.3. Note that condition P43 can be replaced with condition P41 in Lemma 7.3.2 if  > 0 and the corresponding asymptotic expansion in condition .;kC1/ P43 can be omitted in Lemma 7.3.2 for n D 0 if  D 0. .;k/

Remark 7.3.4. According to Lemma 7.3.2, condition P41 .;kC1/ . rem 7.3.2 with condition P43

can be replaced in Theo-

7.3 Asymptotic expansions in Cram´er–Lundberg approximation

465

.;kC1/

is more convenient to work with in applications, since it is forCondition P43 mulated for the claim distributions in terms of the usual moment generating functions ."/ ./ and its derivatives ."/ Œ; n. In many cases there are explicit formulas for ."/ ./. They can effectively be used if the perturbation conditions are given in terms of the parameters of these distribution. For example, let G ."/ .u/ be a gamma-distribution with parameters ."/ and , i.e., ."/

./ D

. ."/ / ; . ."/  /

 < ."/ ;

(7.3.33)

and ."/

Œ; n D a.; n/

. ."/ / ; . ."/  /Cn

 < ."/ ;

(7.3.34)

Q where a.; n/ D nlD1 . C n  l/. Let us assume that the following perturbation condition holds: .kC1/ : ."/ D .0/ C 1 " C    C kC1 "kC1 C o."kC1 /, where 0 < .0/ < 1 and P44 j r j < 1, r D 1; : : : ; k C 1.

As usual we will denote 0 D .0/ . Let us fix some 0 < ˇ < .0/ . It is obvious that every function ."/ Œ; n, n D 0; 1; : : :, has an asymptotic .0; k C 1/-expansion. We will not try to get a general formula but just give explicit expressions for the first two terms in the corresponding asymptotic expansions for the non-trivial case  ¤ 1. Thus, let k  1. By expanding the function . ."/ / in a power series at the point " D 0, we get that (a) . ."/ / D . 0 / C ". 0 /1 1 C "2 . 12 .  1/ 02 . 1 /2 C . 0 /1 2 / C o."2 /. Similarly, by expanding the function . ."/  /Cn in a power series at the point " D 0, we obtain (b) . ."/  /Cn D . 0  /Cn C ". C n/. 0  /Cn1 1 C "2 . 12 . C n/. C n  1/. 0  /Cn2 . 1 /2 C . 0  /1 2 / C o."2 /. Now the corresponding asymptotic .0; 2/-expansions for the functions ."/ Œ; n can be obtained by gathering and equating the coefficients at powers of " in the asymptotic identity .uŒ; 0; n C "uŒ; 1; n C "2 uŒ; 2; n C o."2 //  .. 0  /Cn C ". C n/. 0  /Cn1 1 1 C "2 . C n/. C n  1/. 0  /Cn2 . 1 /2 2  C . 0  /1 2 C o."2 // D a.; n/.. 0 / C ". 0 /1 1 1  C "2 .  1/ 02 . 1 /2 C . 0 /1 2 C o."2 //: 2

(7.3.35)

466

7

Nonlinearly perturbed risk processes

This algorithm yields the following formulas: 1 a.; n/. 0 / ; . 0  /Cn 1 uŒ; 1; n D . 0 /1 1  uŒ; 0; n. C n/. 0  /Cn1 1 ; . 0  /Cn 1 1 .  1/ 02 . 1 /2 C . 0 /1 2 uŒ; 2; n D . 0  /Cn 2 1  uŒ; 0; n . C n/. C n  1/. 0  /Cn2 . 1 /2 2   1 C . 0  / 2  uŒ; 1; n. C n/. 0  /Cn1 1 : (7.3.36) uŒ; 0; n D

Conditions of Theorems 7.3.2 and 7.3.3 become simpler in a model where the claim distribution G ."/ .u/  G .0/ .u/ does not depend on the perturbation parameter ". In this case, condition C22 reduces to condition C21 . Condition D30 automatically holds. .;k/ .;kQ / Conditions P41 and P42 also automatically hold for any k D 0; 1; : : : . Moreover, in this case, vŒ; 0; n D .0/ Œ; n and vŒ; r; n D 0 for r D 1; 2; : : :, n D 0; 1; : : : . This permits to write formulas (7.3.11) for the coefficients bŒ; r; n, given in Lemma 7.3.1, in a simpler form for r; n D 0; 1; : : :, bŒ; r; n D hr vŒ; 0; n;

(7.3.37)

  m X eq hmq : hm D d01 em 

(7.3.38)

where, as in Lemma 7.3.1,

qD1

In some cases, it is convenient to express the condition for the claim distributions directly in terms of the distributions G ."/ .u/. An example of this is a model of conQ taminated claim distributions, G ."/ .u/ D .1  "/G .0/ .u/ C "G.u/. Here " serves as Q the probability of contamination and G.u/ is the contamination distribution. The conQ tamination formula for G ."/ .u/ can be rewritten as G ."/ .u/ D G .0/ .u/ C ".G.u/  .0/ G .u//. This representation suggests the following form of the perturbation condition for the claim distributions: .k/ ."/ P45 : G ."/ .u/ D G .0/ .u/ C G1 .u/" C    C Gk .u/"k C GkC1 .u/o."k /, where Gl .u/, ."/

l D 1; : : : ; k, and GkC1 .u/ are functions (in u) that are right continuous and bounded by a constant that does not depend on ".

7.4 Approximations for ruin probabilities based on asymptotic expansions

467

The following condition replaces condition C22 : C23 : There exists ı > 0 such that: (a) lim0"!0 ' ."/ .ı/ < 1; R1 .0/ (b) c .0/ ' .0/ .ı/ D ˛ .0/ 0 e ıs GN .0/ .ds/ 2 .1; 1/; R1 (c) 0 e ıs jGl .s/j ds < 1, l D 1; : : : ; k; R1 ."/ .s/j ds < 1. (d) lim"!0 0 e ıs jGkC1 .k/

It is readily seen that conditions C23 and P45 imply that condition C22 holds and .;k/ conditions P42 is satisfied for any  < ı. .k/ The coefficients in the expansion given in condition P45 can be calculated with the use of the following formulas: Z 1 vŒ; 0; n D s n e s .1  G .0/ .s// ds; n D 0; 1; : : : ; (7.3.39) 0

and

Z vŒ; r; n D 

1 0

s n e s Gl .s/ ds;

r D 1; 2; : : : ;

n D 0; 1; : : : :

(7.3.40)

7.4 Approximations for ruin probabilities based on asymptotic expansions In this section we give approximations for ruin probabilities based on asymptotic expansions.

7.4.1 Asymptotic expansions for ruin renewal limits .r/

One can weaken the balancing condition B11 used in the Theorem 7.3.2 and replace it with the following: .r/

B12 : u."/ ! 1 in such a way that lim"!0 "r u."/ D %r 2 Œ0; 1/. .k/

.k/

..0/ ;k/

.r/

Under conditions H1 , D30 , C22 , P39 , P40 , P41 , and B12 , we have the following asymptotic relation that is equivalent to the asymptotic relation (7.3.17): P ."/ .u."/ / e

."/ r u."/

 .0/ ..0/ /

! 1 as " ! 0;

(7.4.1)

where r."/ D .0/ C a1 " C    C ar "r ;

(7.4.2)

468

7

Nonlinearly perturbed risk processes

and  ."/ .."/ / is the renewal limit that corresponds to the renewal equation (7.2.3) and is given by formula (7.3.3), i.e., R 1 ."/ s e .1  GN ."/ .s// ds ."/ ."/  . / D 0 R 1 ."/ s G N ."/ .ds/ 0 se R 1 ."/ s R 1 e . .1  G ."/ .u// du/ ds : D 0 R 1 s."/  s .1  G ."/ .s/ ds 0 se ."/ ."/

The asymptotic relation (7.4.1) suggests that we can use the expression e r u   .0/ ..0/ / as an approximation for the ruin probability P ."/ .u."/ /. ."/ ."/ The approximation e r u  .0/ ..0/ / is obtained by replacing the Lundberg ex."/ ponent ."/ with its approximation r calculated using from the asymptotic expansion resulted from applying Theorem 7.3.2 to the renewal equation (7.2.3), ."/ D .0/ C a1 " C    C ak "k C o."k /:

(7.4.3)

."/ ."/

The approximation e r u  .0/ ..0/ / also contains the limit constant  .0/ ..0/ / given by formula (7.3.3). Theorem 1.4.4, applied to the renewal equation (7.2.3), shows that  ."/ .."/ / !  .0/ ..0/ / as " ! 0:

(7.4.4)

Indeed, as was shown in the proof of Theorem 7.3.2, conditions H1 , D30 and C22 , .k/ ..0/ ;k/ P40 , and P41 imply that conditions D7 , C2 and F6 hold in the case where  > 0 or D6 , C1 , and F5 if .0/ D 0. Thus, Theorem 1.4.4 can be applied to the renewal equation (7.2.3) to give relation (7.4.4). On the other hand, conditions H1 , D30 , and C22 imply that there exists "0 > 0 such ."/ that (a) condition C21 holds for the risk process u .t /, t  0, for every "  "0 . Thus, for every "  "0 , the classical Cram´er–Lundberg approximation for ruin probabilities takes place, .k/ P39 , .0/

P ."/ .u/  e 

."/ u

 ."/ .."/ / as u ! 1: ."/ ."/

(7.4.5)

This shows that the approximation e r u  .0/ ..0/ / could be improved by replacing the coefficient  .0/ ..0/ / with  ."/ .."/ /. Moreover, it also seems logical to replace  ."/ .."/ / with some approximation using the expansion of this coefficient in an asymptotic power series with respect to the perturbation parameter ". Such expansions can be obtained by applying Theorem 2.3.2 to the renewal equation (7.2.3). Introduce the mixed power-exponential moment generating functions by Z 1  Z 1 ! ."/ ./ D e s .1  G ."/ .u// du ds 0

D ."/

s

Z

1 0

e s .1  GN ."/ .s// ds;

469

7.4 Approximations for ruin probabilities based on asymptotic expansions

and the mixed power-exponential moment generating functions for n D 0; 1; : : :, Z 1  Z 1 ! ."/ Œ; n D s n e s .1  G ."/ .u// du ds 0

D ."/

s

Z

1 0

s n e s .1  GN ."/ .s// ds:

By the definition, ! ."/ Œ; 0 D ! ."/ ./. Choose an arbitrary .0/ < ˇ < ı. Condition C22 (a) implies that (b) ' ."/ .ˇ/ < 1 for "  "2 . On the other hand, integration by parts shows that (c) ! ."/ .ˇ/ D ˇ 1 .' ."/ .ˇ/  1/. Using (c), we get for "  "2 and n D 0; 1; : : : that ! ."/ Œˇ; n  cn ! ."/ .ˇ/ < 1;

(7.4.6)

where cn D cn .ı; ˇ/ D sups0 s n e .ıˇ /s < 1. Also note that for   ˇ, ! ."/ Œ; n is the n-order derivative of order n of the function ! ."/ ./. Let N 1."/ be a random variable with the distribution function GN ."/ .s/. The following relation links the moment generating functions ! ."/ ./ and ' ."/ ./: ! ."/ ./ D ."/ ( D ( D

Z

1

0

e s .1  GN ."/ .s// ds D ."/ E  N 1."/



."/

.Ee



."/

."/ E N 1

 1/=

Z

."/

N 1 0

e s ds

for  ¤ 0, for  D 0,

.' ."/ Œ; 0  ' ."/ Œ0; 0/= ' ."/ Œ0; 1

for  ¤ 0, for  D 0.

(7.4.7)

Note that we have used the identity ."/ D ' ."/ Œ0; 0 in relation (7.4.7). Relation (7.4.7) yields the following recurrence relations for n D 1; 2; : : :, "  "2 , and   ˇ: ( .' ."/ Œ; n  n! ."/ Œ; n  1/= for  ¤ 0;   ˇ, ."/ (7.4.8) ! Œ; n D ' ."/ Œ0; n C 1=.n C 1/ for  D 0. Relation (7.4.8) can be obtained similarly to relation (7.3.29)). Relation (a) ' ."/ Œ; 0 D ! ."/ Œ; 0 C ."/ ,   ˇ, follows from formula (7.4.7). By differentiating (a) we get the following relation: (b) ' ."/ Œ; n D ! ."/ Œ; n C n! ."/Œ; n  1, n D 1; 2; : : :,   ˇ. In particular, (b) yields the following relation for  D 0: (c) ' ."/ Œ0; n D n! ."/ Œ0; n  1, n D 1; 2; : : :,   ˇ. Relations (a)–(c) imply relation (7.4.8).

470

7

Nonlinearly perturbed risk processes .;kC1/

Let us formulate a perturbation condition implied by condition P41

,

.;k/ : ! ."/ Œ; n D ! .0/ Œ; n C wŒ; 1; n" C    C wŒ; k  n; n"kn C o."kn /, P46 where jwr Œ; nj < 1, r D 1; : : : ; k  n, n D 0; : : : ; k. It also is convenient to define wŒ; 0; n D ! .0/ Œ; n, n D 0; : : : ; k. The following lemma is an obvious corollary of relations (7.4.7) and (7.4.8). .;kC1/

Lemma 7.4.1. Let condition H1 , C22 hold, and condition P41 be satisfied for .;k/ some   ˇ. Then condition P46 holds for   ˇ, and the asymptotic expansions in this condition take the following form for n D 0; : : : ; k: ! ."/ Œ; n D ! .0/ Œ; n C wŒ; 1; n" C    C wŒ; k  n; n"kn C o."kn /;

(7.4.9)

where the coefficients wŒ; r; n, r D 0; : : : ; k  n, n D 0; : : : ; k, are given for r D 0; : : : ; k, n D 0 by ( .vŒ; r; 0  vŒ0; r; 0/= if  ¤ 0, wŒ; r; 0 D (7.4.10) vŒ0; r; 1 if  D 0, and, for r D 0; : : : ; k  n, n D 1; : : : ; k, by ( .vŒ; r; n  nwŒ; r; n  1/= wŒ; r; n D vŒ0; r; n C 1=.n C 1/

if  ¤ 0, if  D 0.

(7.4.11)

.;kC1/ Remark 7.4.1. Note that condition P41 can be replaced in Lemma 7.4.1 with .;k/ condition P32 if  > 0, and the corresponding asymptotic expansion in condition .;kC1/ P41 can be omitted in Lemma 7.3.2 for n D 0 if  D 0. .;kC1/

Remark 7.4.2. According to Lemma 7.3.2, condition P41 .;kC2/ . P43

is implied by condition

The following theorem gives asymptotic expansions for the renewal limits  .."/ /. This is a corollary of Theorem 2.3.2 applied to the renewal equation (7.2.3). ..0/ ;kC2/ Note that we use condition P43 which, according to Lemmas 7.3.2 and 7.4.1, ."/

..0/ ;kC1/

implies that both conditions P46

..0/ ;k/

and P41

hold.

.k/ .k/ Theorem 7.4.1. Let conditions H1 D30 , C22 , P39 , P40 , and P. 43 the functional  ."/ has the asymptotic expansions

 ."/ .."/ / D

.0/ ;kC2/

hold. Then

! .0/ Œ.0/ ; 0 C f10 " C    C fk0 "k C o."k /

' .0/ Œ.0/ ; 1 C f100 " C    C fk00 "k C o."k /

D  .0/ ..0/ / C f1 " C    C fk "k C o."k /;

(7.4.12)

471

7.4 Approximations for ruin probabilities based on asymptotic expansions

where the coefficients fn0 ; fn00 are given by the formulas f00 D ! .0/ Œ.0/ ; 0 D wŒ.0/ ; 0; 0, f10 D wŒ.0/ ; 1; 0 C wŒ.0/ ; 0; 1a1 , f000 D ' .0/ Œ.0/ ; 1 D vŒ.0/ ; 0; 1, f100 D vŒ.0/ ; 1; 1 C vŒ.0/ ; 0; 2a1 , and, for n D 0; : : : ; k, fn0

D wŒ

.0/

; n; 0 C

n X

wŒ.0/ ; n  q; 1aq

qD1

C

X

n X

wŒ

.0/

; n  q; m 

2mn qDm

q1 Y

X

n

app =np Š;

(7.4.13)

n1 ;:::;nq1 2Dm;q pD1

and fn00 D vŒ.0/ ; n; 1 C

n X

vŒ.0/ ; n  q; 2aq

qD1

X

C

n X

vŒ

.0/

; n  q; m C 1 

2mn qDm

q1 Y

X

n

app =np Š; (7.4.14)

n1 ;:::;nq1 2Dm;q pD1

and the coefficients fn are given by the recurrence formulas f0 D  .0/ ..0/ / D f00 =f000 and, for n D 0; : : : ; k,  fn D

fn0



n1 X

00 fnq fq



=f000 :

(7.4.15)

qD0

Proof. We shall apply Theorem 2.3.2 to the renewal equation (7.2.3). As usual, we can take .0/ < ˇ < ı. .k/ .k/ , P40 , and P. Conditions H1 , D30 and C22 , P39 41

tions D11 , C2 hold. Conditions and P. 46

.0/ ;k/

.0/ ;k/

..0/ ;kC2/ E1 , C22 , P43

obviously imply that condi..0/ ;kC1/

imply that conditions P41

hold. This follows from Lemmas 7.3.2 and 7.4.1. Condition P3.kC1/ takes ..0/ ;kC1/

.k/

..0/ ;k/

the form of condition P41 , and condition P7 becomes condition P46 . Fi.k/ nally, as was shown in the proof of Theorem 7.3.2, conditions H1 , D30 and C22 , P39 , .k/

..0/ ;k/

imply that condition F6 also holds. P40 , and P41 Thus, we apply Theorem 2.3.2 to the renewal equation (7.2.3) and obtain the asymptotic relations (7.4.12). Remark 7.4.3. The coefficients f10 , f100 , f1 and f20 , f200 , f2 can be calculated with the use of formulas (2.3.24) given in Subsection 2.2.1, where the coefficients br;n and cr;n should be replaced, respectively, with the coefficients vŒ.0/ ; r; n and wŒ.0/ ; r; n,

472

7

Nonlinearly perturbed risk processes

i.e., f00 D wŒ.0/ ; 0; 0; f10 D wŒ.0/ ; 1; 0 C wŒ.0/ ; 0; 1a1 wŒ.0/ ; 0; 1vŒ.0/ ; 1; 0 ; vŒ.0/ ; 0; 1

D wŒ.0/ ; 1; 0 

f20 D wŒ.0/ ; 2; 0 C wŒ.0/ ; 0; 1a2 C wŒ.0/ ; 1; 1a1 C

wŒ.0/ ; 0; 2a12 ; 2

f000 D vŒ.0/ ; 0; 1; f100 D vŒ.0/ ; 1; 1 C vŒ.0/ ; 0; 2a1 D vŒ.0/ ; 1; 1 

vŒ.0/ ; 0; 2vŒ.0/ ; 1; 0 ; vŒ.0/ ; 0; 1

f200 D vŒ.0/ ; 2; 1 C vŒ.0/ ; 0; 2a2 C vŒ.0/ ; 1; 2a1 C

vŒ.0/ ; 0; 3a12 ; 2

f0 D

f00 wŒ.0/ ; 0; 0 D ; f000 vŒ.0/ ; 0; 1

f1 D

1 wŒ.0/ ; 1; 0 wŒ.0/ ; 0; 1vŒ.0/ ; 1; 0 0 00 ;  00 .f1  f0 f1 / D f0 vŒ.0/ ; 0; 1 vŒ.0/ ; 0; 12

wŒ.0/ ; 0; 0vŒ.0/ ; 0; 2vŒ.0/ ; 1; 0 wŒ.0/ ; 0; 0vŒ.0/ ; 1; 1 C ; vŒ.0/ ; 0; 12 vŒ.0/ ; 0; 13 1 f2 D 00 .f20  f0 f200  f1 f100 /: (7.4.16) f0 

7.4.2 Approximations for ruin probabilities The asymptotic expansions (7.4.12) can be used to construct approximations for the coefficients  ."/ .."/ /. We can use the following approximation formula for  ."/ .."/ /, based on the asymptotic expansion (7.4.12): ."/

l

D  .0/ ..0/ / C f1 " C    C fl "l :

."/

(7.4.17)

."/

Approximations r D .0/ C a1 " C    C ar "r and l D  .0/ ..0/ / C f1 " C    C fl "l for the Lundberg exponent ."/ and the coefficients  ."/ .."/ /, respectively, leads to the following approximation for the ruin probabilities: ."/

."/

Pr;l .u/ D e r

u

."/

l :

(7.4.18)

By using the parameters r and l one can control the highest order of moments of the claim distribution involved in the approximation formulas.

7.5 Asymptotic expansions for the surplus prior to and at the time of a ruin

473

."/

Approximation Pr;l .u/ is asymptotically equivalent, under conditions of Theorem .r/

7.4.1 and condition B12 , to the initial approximation given by the Cram´er–Lundberg ."/ approximation for the ruin probability P ."/ .u/ by the quantity PQ ."/ .u/ D e  u  ."/  ."/ .."/ / or by its variant PO ."/ .u/ D e  u  .0/ ..0/ /, i.e., P ."/ .u."/ / ."/

Pr;l .u."/ /



P ."/ .u."/ / P ."/ .u."/ /  ! 1 as " ! 0: PQ ."/ .u."/ / PO ."/ .u."/ /

(7.4.19)

All the approximation considered above give a zero asymptotic relative error for the ruin probabilities both in models where the limit Lundberg exponent is .0/ > 0, which corresponds to the case of the Cram´er–Lundberg approximation for perturbed risk processes, and in a model where .0/ D 0, which corresponds to the case the diffusion approximation.

7.5 Asymptotic expansions for the distribution of the surplus prior to and at the time of a ruin In this section, we consider an additional example that shows how to apply the methods of asymptotic analysis of a perturbed renewal equation to perturbed risk processes. We obtain asymptotic expansions for the distribution of the surplus prior to and at the time of a ruin.

7.5.1 The surplus prior to and at the time of a ruin Let us introduce, for u  0, a random variable, which is the time of a ruin,

u."/ D inf.t  0 W u."/ .t / < 0/: This random variable takes values in the interval .0; 1. The ruin probability can ."/ be expressed with the use of this random variable as P ."/ .u/ D Pf u < 1g. Let us now introduce, for u; x; y  0, the random variables u."/ . u."/  0/ and ."/ ."/  u . u / that are, respectively, the surplus prior to and at the time of the ruin. These ."/ random variables are well defined if u < 1. Let us assign them value C1 if ."/

u D 1. Let us now introduce the surplus probabilities that define the joint distribution of the surpluses prior to and at the time of the ruin, ."/ Px;y .u/ D Pf u."/ < 1;  u."/ . u."/ / > x; u."/ . u."/  0/ > yg;

x; y  0:

474

7

Nonlinearly perturbed risk processes

A starting point in our asymptotic analysis is the following renewal equation for the ."/ .u/; u  0 given in Schmidli (1999): surplus probabilities Px;y ."/ Px;y .u/ D ˛ ."/ .1  GN ."/ .u _ y C x// Z u ."/ ."/ C˛ Px;y .u  s/GN ."/ .ds/; 0

u  0:

(7.5.1)

This renewal equation should be compared with the renewal equation (7.2.3) satisfied by the ruin probabilities P ."/ .u/; u  0. Both equations have the same generating distribution F ."/ .s/ D ˛ ."/ GN ."/ .s/ and differ only in the forcing functions. In the case of the renewal equation (7.5.1), the forcing function is q ."/ .s/ D ˛ ."/ .1  GN ."/ .u _ y C x//, while in the case of the renewal equation (7.2.3), the forcing function is given by q ."/ .s/ D ˛ ."/ .1  GN ."/ .u//. Note that, by the definition, both random variables, the surplus prior to the ruin, ."/ ."/ ."/ ."/ u . u  0/, and at the time of the ruin,  u . u /, are positive random variables ."/ with probability 1. This implies that P0;0 .u/ D P ."/ .u/; u  0.

7.5.2 Diffusion approximation for the surplus prior to and at the time of a ruin for perturbed risk processes For x; y  0, denote

R1 ."/ x;y

D

0

.1  GN ."/ .u _ y C x// du R1 : N ."/ 0 uG .du/

(7.5.2)

R1 Conditions H1 and M16 imply that both integrals are finite, 0 .1  GN ."/ .u _ y C R1 ."/ x// du < 1 and 0 uGN ."/ .du/ < 1, and therefore, the quantity x;y is well defined for all " small enough. ."/ ."/ By the definition, (a) Px;y 2 Œ0; 1, (b) Px;y .u/ is a non-increasing function in ."/ x  0 and y  0, and (c) P0;0 .u/ D P ."/ .u/. The following theorem is a variant of the diffusion approximation for perturbed risk processes given in Lemma 7.2.2. Theorem 7.5.1. Let conditions H1 , D30 , M16 , and B10 be satisfied. Then, for x; y  0, ."/ ."/ .0/ %=m N .u / ! x;y e Px;y

.0/

as " ! 0;

(7.5.3)

where m N .0/ D ˇ .0/ =2.0/ . Proof. The proof is absolutely analogous to the proof of Lemma 7.2.2. The only difference is in the formula for the corresponding renewal limit. In the case of the ruin probabilities P" .u/ treated in Lemma 7.2.2, the corresponding renewal limit, given in ."/ (7.2.9), equals 1. In the case of the quantities Px;y .u/, the corresponding renewal limit .0/ is Px;y , which explains the difference between the limits in the asymptotic relations (7.2.10) and (7.5.3).

7.5 Asymptotic expansions for the surplus prior to and at the time of a ruin

475

7.5.3 Generalised Cram´er–Lundberg and diffusion approximations for the surplus prior to and at the time of a ruin for perturbed risk processes Denote

R1

.1  GN ."/ .s _ y C x// ds R1 ."/ s G N ."/ .ds/ 0 se R 1 ."/ s R 1 . s_yCx .1  G ."/ .u// du/ ds 0 e D : (7.5.4) R1 ."/ s .1  G ."/ .s/ ds 0 se R 1 ."/ Conditions H1 , D30 –D32 , and C22 imply that 0 e  s.1  GN ."/ .u _ y C x// du < R1 ."/ ."/ 1 and 0 ue  u GN ."/ .du/ < 1 and, therefore, the quantity x;y .."/ / is well defined for all " small enough. The following theorem is an analogue of Theorem 7.3.1. ."/ .."/ / x;y

D

0

e

."/ s

Theorem 7.5.2. Let conditions H1 , D30 –D32 , and C22 hold. Then, for any 0  u."/ ! 1 as " ! 0, the following asymptotic relation holds for x; y  0: ."/

Px;y .u."/ / .0/ .0/ . / as " ! 0: ! x;y expf."/ u."/ g

(7.5.5)

Proof. The proof repeats the proof of Theorem 7.3.1. The only difference is in the formula for the corresponding renewal limit. In the case of the ruin probabilities P" .u/ considered in Theorem 7.3.1, the corresponding renewal limit given in (7.3.3) ."/ is  .0/ ..0/ /. In the case of the quantities Px;y .u/, the corresponding renewal limit .0/ .0/ is x;y . /, which explains the difference between the limits in the asymptotic relations (7.3.5) and (7.5.5).

7.5.4 Exponential expansion for the distribution of the surplus prior to and at the time of a ruin Finally, let us formulate an analogue of Theorem 7.3.2 for for the distribution of the surplus prior to and at the time of a ruin. Statements (i) and (ii) do not change. Thus we only give an analogue of statement (iii) of the theorem. .k/

.k/

..0/ ;k/

Theorem 7.5.3. Let conditions H1 , D30 , C22, P39 , P40 , and P41 hold and con.r/ dition B11 be satisfied for some 1  r  k. Then the following asymptotic relation holds for x; y  0: ."/ Px;y .u."/ / .0/ .0/ . / as " ! 0; (7.5.6) ! e %r ar x;y expf..0/ C a1 " C    C ar 1 "r 1 /u."/ g

where the coefficients an , n D 1; : : : ; k, are given by formulas (7.3.15).

476

7

Nonlinearly perturbed risk processes

Proof. The proof repeats the proof of Theorem 7.3.2. Again, the only difference is in the formula for the corresponding renewal limit. In the case of the ruin probabilities P" .u/ treated in Theorem 7.3.2, the corresponding renewal limit given in (7.3.3) is ."/  .0/ ..0/ /. In the case of the quantities Px;y .u/, the corresponding renewal limit is .0/ .0/ x;y . /, which explains the difference between the limits in the asymptotic relations (7.3.17) and (7.5.6).

Chapter 8

Supplements

The last Chapter 8 contains three supplements. The first one gives some basic arithmetic operation formulas for scalar and matrix asymptotic expansions. In the second supplement, we discuss some new prospective directions for future research studies in the area and make comments on some already available results. Let us point out some of them. In the book, we study quasi-stationary phenomena for nonlinearly perturbed stochastic systems with continuous time. However, it is clear that similar results can also be obtained for discrete time models which are important by themselves and in applications. Another direction for future studies is the exponential asymptotic expansions, which are based on non-polynomial systems of infinitesimals, in mixed ergodic and large-deviation theorems for nonlinearly perturbed regenerative, Markov, and semi-Markov processes. In the book, we treat models of nonlinearly perturbed Markov chains and semi-Markov processes with a finite phase space. It would be important to realise an analogous program of studies for Markov chains and semiMarkov type processes with countable and general phase spaces. Another direction for advancing the studies carried out in the book is connected with models for nonlinearly perturbed Markov chains and semi-Markov processes with uncoupled phase spaces. Explicit estimates for the remainder terms in the exponential asymptotic expansions for nonlinearly perturbed stochastic systems also make a natural object for future studies. Studies of pseudo- and quasi-stationary phenomena for nonlinearly perturbed queueing and reliability systems, various epidemic models, models of population dynamics, and other biological systems are very important and prospective directions for the future research. Examples given in the book make just the first step in exploring this unlimited area. Quasi-stationary phenomena and related problems were a subject of intensive studies during several decades. The bibliography contained in the book has more than 1000 references. The last supplement contains brief bibliographical remarks on the works in he related areas dealt with in this comprehensive bibliography.

8.1 Arithmetics of asymptotic expansions In this section we give formulas for working with asymptotic expansions. These formulas are used in the book on numerous occasions. They are known and can be found in the existing literature. We give them here to make the book more self-contained.

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8.1.1 Asymptotic expansions Let A."/ be a function that is defined on an interval 0 < "  "0 and given on this interval by an asymptotic expansion, A."/ D ah "h C    C ak "k C o."k /;

(8.1.1)

where (a) 0  h  k < 1 are integers, and (b) the coefficients ah ; : : : ; ak are real numbers. We call such an asymptotic expansion an .h; k/-expansion. In fact, representation (8.1.1) means that A."/ is a sum of a polynomial function of " of the order k with the first h  1 coefficients equal to zero, and a function o."k / satisfying o."k /="k ! 0 as " ! 0. In particular, .0; 0/-expansion has the form A."/ D a0 C o.1/, where o.1/ ! 0 as " ! 0. We say that an .h; k/-expansion A."/ is pivotal if it is known that ah ¤ 0. Consider four asymptotic expansions, A."/ D ahA "hA C    C akA "kA C o."kA /, B."/ D bhB "hB C    C bkB "kB C o."kB /, C."/ D chC "hC C    C ckC "kC C o."kC /, and D."/ D dhD "hD C    C dkD "kD C o."kD /. Lemma 8.1.1. The above asymptotic expansions have the following “operational” rules: (i) If A."/ is an .hA ; kA /-expansion and d is a constant, then dA."/ is an .hA ; kA /expansion with the corresponding coefficients dar ; r D hA ; : : : ; kA . This expansion is pivotal if and only if dahA ¤ 0. (ii) If A."/ is an .hA ; kA /-expansion and B."/ is an .hB ; kB /-expansion, then C."/ D A."/ C B."/ is an .hC ; kC /-expansion with hC D hA ^ hB , kC D kA ^ kB , and the coefficients satisfy chC Cr D ahC Cr C bhC Cr ; r D 0; : : : ; kC  hC , where ahC Cr D 0 for 0  r < hA  hC and bhC Cr D 0 for 0  r < hB  hC . The expansion C."/ is pivotal if and only if chC D ahC C bhC ¤ 0. (iii) If A."/ is an .hA ; kA /-expansion and B."/ is an .hB ; kB /-expansion, then C."/ D A."/  B."/ is an .hC ; kC /-expansion with hC D hA C hB , k C D .hA C kB / ^ .kA C hB /, and the coefficients are given by chC Cr D P r D 0; : : : ; kC  hC : The expansion C."/ is pivotal 0ir ahA Ci bhB Cr i ; if and only if chC D ahA bhB ¤ 0. (iv) If B."/ is a pivotal .hB ; kB /-expansion, then C."/ D "hB =B."/ is a pivotal .0; kC /-expansion with kC D kB  hB and the coefficients satisfy c0 D bh1 B P and cr D bh1 b c , r D 1; : : : ; k . r i C h Ci 1ir B B (v) If A."/ is an .hA ; kA /-expansion, B."/ is a pivotal .hB ; kB /-expansion, then D."/ D "hB A."/=B."/ is an .hP D ; kD /-expansion with hD D hA , kD D kA ^ .hA CkB hB /, and dhD Cr D 0ir ci ahA Cri , r D 0; : : : ; kD hD , where cj ; j D 0; : : : ; kC are coefficients of the .0; kC /-expansion C."/ D "hB =B."/

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given in (iv). The expansion D."/ is pivotal if and only if dhD D ahA c0 D ahA =bhB ¤ 0. (vi) If A."/ is an .hA ; kA /-expansion, B."/ is a pivotal .hB ; kB /-expansion, then D."/ D "hB A."/=B."/ is an .hD ; kD /-expansion with hD D hA , kD D kA ^ .hA C kB  hB /, and the coefficients satisfying dhD Cr D bh1 .ahA Cr  B P 1ir bhB Ci dhD Cr i /, r D 0; : : : ; kD  hD . The expansion D."/ is pivotal if and only if dhD D ahA =bhB ¤ 0.

8.1.2 Asymptotic matrix expansions Lemma 8.1.1 can be generalised to a model of asymptotic matrix expansions. Let A."/ D jjAij ."/jj be an m  n matrix-valued function that is defined on an interval 0 < "  "0 and represented on this interval by the asymptotic matrix expansion A."/ D Ah "h C    C Ak "k C o."k /;

(8.1.2)

where (a) 0  h  k < 1 are integers, and (b) the matrix coefficients Ah D jjah;ij jj; : : : ; Ak D jjak;ij jj are m  n matrices with real-valued elements, and (c) o."k / D jjoij ."k /jj is an m  n matrix-valued function of " such that oij ."k /="k ! 0 as " ! 0, i D 1; : : : ; m; j D 1; : : : ; n. We call such an asymptotic expansion an .m; n/-matrix .h; k/-expansion. Let us now consider four asymptotic matrix expansions A."/ D AhA "hA C    C AkA "kA Co."kA /, B."/ D BhB "hB C    C BkB "kB C o."kB /, C."/ D ChC "hC C    C CkC "kC C o."kC /, and D."/ D DhD "hD C    C DkD "kD C o."kD /. The number of lines and columns in the corresponding matrices will be specified in each particular case. Lemma 8.1.2. The above asymptotic matrix expansions have the following have the following “operational” rules: (i) If A."/ is an .m; n/-matrix .hA ; kA /-expansion and d is a constant, then d A."/ is an .m; n/-matrix .hA ; kA /-expansion with the corresponding matrix coefficients d Ar , r D hA ; : : : ; kA . (ii) If A."/ is an .m; n/-matrix .hA ; kA /-expansion, B."/ is an .m; n/-matrix .hB ; kB /-expansion, then C."/ D A."/ C B."/ is an .m; n/-matrix .hC ; kC /expansion with hC D hA ^ hB , kC D kA ^ kB , ChC Cr D AhC Cr C BhC Cr , r D 0; : : : ; kC  hC , where AhC Cr D 0 for 0  r < hA  hC and BhC Cr D 0 for 0  r < hB  hC , where 0 is a matrix with all elements equal to zero. (iii) If A."/ is an .m; l/-matrix .hA ; kA /-expansion and B."/ is an .l; n/-matrix .hB ; kB /-expansion, then C."/ D A."/  B."/ is an .m; n/-matrix .hC ; kC /expansion with hC D hAP ChB , kC D .hA CkB /^.kA ChB /, and the coefficients are given by ChC Cr D 0ir AhA Ci BhB Cri , r D 0; : : : ; kC  hC .

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(iv) If B."/ is an .n; n/-matrix .hB ; kB /-expansion and det.Bh / ¤ 0, then C."/ D "hB B."/1 is an .n; n/-matrix P .0; kC /-expansion with kC D kB  hB , and 1 1 C0 D BhB and Cr D BhB 1ir BhB Ci Cri , r D 1; : : : ; kC . (v) If A."/ is an .n; m/-matrix .hA ; kA /-expansion, B."/ is an .n; n/-matrix .hB ; kB /-expansion, and det.BhB / ¤ 0, then D."/ D "hB B."/1 A."/ is an .n; m/-matrix .hP C ; kC /-expansion with hD D hA , kD D kA ^ .hA C kB  hB /, DhD Cr D 0ir Ci AhA Cr i , r D 0; : : : ; kD  hD , where Cj ; j D 0; : : : ; kC , are coefficients of the .n; n/-matrix .0; kC /-expansion C."/ D "hB B."/1 given in (iv). (vi) If A."/ is an .n; m/-matrix .hA ; kA /-expansion, B."/ is an .n; n/-matrix .hB ; kB /-expansion, det.BhB / ¤ 0, then D."/ D "hB B."/1 A."/ is an .n; m/matrix .hD ; kD /-expansion Pwith hD D hA , kD D kA ^ .hA C kB  hB /, and .A  DhD Cr D B1 h Cr 1ir BhB Ci Dhd Cri /, r D 0; : : : ; kD  hD . A hB

8.1.3 Nonlinearly perturbed systems of linear equations Let B."/ D jjBij ."/jj be an n  n matrix-valued function and a."/ D jjai ."/jj be an n  1 matrix-valued (column vector-valued) function with real-valued elements that are defined in an interval 0 < "  "0 . We consider a system of linear equations B."/x."/ D a."/:

(8.1.3)

A solution of the system x."/ D jjxi ."/jj is sought for in the class of (n  1)matrix (column vector-valued) functions with real-valued elements that are defined on an interval 0 < "  "00 , where 0 < "00 < "0 . We assume that a."/ and B."/ can be represented in the form of asymptotic matrix expansions, a."/ D aha "ha C    C aka "ka C o."ka / with n  1 matrix-valued (vectorvalued) coefficients ar D jjar;i jj; r D ha ; : : : ; ka , and B."/ D BhB "hB C  CBkB "kB Co."kB / with n  n matrix-valued coefficients Br D jjbr;ij jj; r D hB ; : : : ; kB . The following lemma gives conditions for system (8.1.3) to have a unique solution that can be represented in the form of an asymptotic expansion. Lemma 8.1.3. Let a."/ be an .n; 1/-matrix .ha ; ka /-expansion, B."/ an .n; n/-matrix .hB ; kB /-expansion, and det.BhB / ¤ 0. Then we have the following: (i) xQ ."/ D "hB B."/1 a."/ is an .n; 1/-matrix (vector) .hx ; kx /-expansion with hx D ha ; kx D ka ^ .ha C kB  hB /, and xQ hx Cr D B1 hB .aha Cr 

X 1ir

BhB Ci xQ hx Cri /;

r D 0; : : : ; kx  hx :

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481

(ii) xQ ."/ D "hB B."/1 a."/ is an .n; 1/-matrix (vector) P .hx ; kx /-expansion with hx D ha ; kx D ka ^ .ha C kB  hB /, and xQ hx Cr D 0ir Ci aha Cri ; r D 0; : : : ; kx  hx , where Cj ; j D 0; : : : ; kC , are coefficients of the .n; n/-matrix .0; kC /-expansion C."/ D "hB B."/1 given in item (iv) of Lemma 8.1.2, i.e., and defined by the recurrence formulas C0 D B1 hB X BhB Ci Cri ; r D 1; : : : ; kC : Cr D B1 hB 1ir

Proof. Obviously, "hB B."/ ! BhB as " ! 0. Therefore, the matrix B."/ has nonzero determinant for all positive " small enough, say 0 < "  "00 < "0 . Thus, for such ", there exists a unique solution of the system of linear equations (8.1.3), which is given by the formula x."/ D B."/1 a."/. Now, the proof follows from parts (v) and (vi) of Lemma 8.1.2. Remark 8.1.1. It is useful to note that xQ ."/ D "hB x."/, where x."/ D B."/1 a."/ is a unique solution of the system of linear equations (8.1.3). Let B."/ D jjBij ."/jj be an n  n matrix-valued function and a."/ D jjai ."/jj an n  1 matrix-valued (column vector-valued) function with real-valued elements that are defined on an interval 0 < "  "0 . We consider a system of linear equations given in a form that often appears in applications to stochastic systems, x."/ D a."/ C A."/x."/:

(8.1.4)

As before, we assume that a."/ and A."/ can be represented in the form of asymptotic matrix expansions, a."/ D aha "ha C  Caka "ka Co."ka / with n1 matrix-valued (vector-valued) coefficients ar D jjar;i jj; r D ha ; : : : ; ka , and A."/ D AhA "hA C   C AkA "kA Co."kA / with nn matrix-valued coefficients Ar D jjar;ij jj; r D hA ; : : : ; kA . Below, we denote by 0 the n  n matrix with all zero elements and by I D jjıi;j jj a unit n  n matrix with zero non-diagonal and unit diagonal elements. Let us also introduce a norm of an n  n matrix A by X jAj D max jaij j: 1in

1j n

Lemma 8.1.4. Let a."/ be an .n; 1/-matrix .ha ; ka /-expansion, A."/ an .n; n/-matrix .0; kA /-expansion, and jAn0 j ! 0 as n ! 1. Then the following holds: (i) x."/ D ŒI  A."/1 a."/ is an .n; 1/-matrix .hx ; kx /-expansion P with ka ^ hx D ha , kx D .ha C kA /, and xhx Cr D ŒI  A0 1 .aha Cr C 1ir Ai xhx Cri /, r D 0; : : :, kx  hx . (ii) x."/ D ŒI  A."/1 a."/ is an .n; 1/-matrix P.hx ; kx /-expansion with hx D ha , kx D ka ^ .ha C kA /, and xhx Cr D 0ir Ci aha Cri , r D 0; : : : ;

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kx  hx , where Cj ; j D 0; : : : ; kC D kA , are coefficients of the .n; n/-matrix .0; kC /-expansion C."/ D ŒI  A."/1 P defined by the recurrence formulas 1 1 C0 D ŒI  A0  and Cr D ŒI  A0  1ir Ci Ari ; r D 1; : : : ; kC . Proof. We employ two basic inequalities for the matrix norms, jABj  jAjjBj and jA C Bj  jAj C jBj. Since jAn0 j ! 0 as n ! 1, there exists m such that jAm 0j < m1 a < 1. Let also b D max.jA0 j; : : : ; jA0 j/. Obviously, A."/ ! A.0/ as " ! 0 and, therefore, jA."/m j ! jA.0/m j as " ! 0, and b" D max.jA."/j; : : : ; jA."/m1 j/ ! b as " ! 0. Thus, jA."/jn  .b C 1/aŒn=m , n D 0; 1; : : :, for all positive " small enough, say 0 < " < "00 . This implies that sup""0 jA."/jn ! 0 as n ! 1. The 0 last asymptotic relation implies that there exists an inverse matrix, .I  A."//1 D I C A."/ C A."/2 C    , for every 0 < " < "00 . For such ", there exists a unique solution of the system of linear equations (8.1.3), which is given by the formula x."/ D .I  A."//1 a."/. Now, the proof follows from Lemma 8.1.3 applied to the case where B."/ D .I  A."//. Lemma 8.1.5. Let a."/ be an .n; 1/-matrix .ha ; ka /-expansion, A."/ an .n; n/-matrix .hA ; kA /-expansion, and hA > 0. Then (i) x."/ D ŒI  A."/1 a."/ is an .n; 1/-matrix .hx ; kx /-expansion with hx D ha , P kx D .ha CkA /^ka , and xhx Cr D ŒIA0 1 .aha Cr C hA ir Ai xhx Cri /, r D 0; : : : ; kx  hx . (ii) x."/ D ŒI  A."/1 a."/ is an .n; 1/-matrix .hx ; kx /-expansion with hx D ha , P kx D .ha C kA / ^ ka , and xhx Cr D 0ir Ci aha Cri , r D 0; : : : ; kx  hx , where Cj , j D 0; : : : ; kC are coefficients of the .n; n/-matrix .0; kC /-expansion 1 C."/ D ŒI  A."/P defined by the recurrence formulas C0 D ŒI  A0 1 and 1 Cr D ŒI  A0  hA ir Ci Ar i , r D 1; : : : ; kC . Proof. Lemma 8.1.5 is a corollary of Lemma 8.1.4. Indeed, the matrix .hx ; kx /expansion A."/ D AhA "hA C    C AkA "kA C o."kA / can be represented in the form of a matrix .0; kx /-expansion as A."/ D 0 C 0"C   C 0"hA 1 CAhA "hA C  CAkA "kA C o."kA /. In this case, A0 D 0, thus condition, An0 ! 0 as " ! 0, holds. The asymptotic matrix expansions for x."/ given in Lemma 8.1.4 reduce to the corresponding expansions given in Lemma 8.1.5.

8.2 New directions for the research This book is devoted to studies of quasi-stationary phenomena in nonlinearly perturbed stochastic processes and systems. Asymptotic results presented in the book include the following: (i) mixed ergodic theorems (for the state of the process or the stochastic system) and limit theorems (for lifetimes) which describe transition phenomena; (ii) mixed ergodic and large deviation theorems that describe pseudo- and quasi-stationary

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483

phenomena; (iii) exponential expansions in mixed ergodic and large deviation theorems; (iv) theorems on convergence of quasi-stationary distributions; and (v) asymptotic expansions for quasi-stationary distributions. The used methods are based on exponential asymptotics for a nonlinearly perturbed renewal equation. The classes of processes for which this program is realised include nonlinearly perturbed regenerative processes, semi-Markov processes, and continuous time Markov chains with absorption. Applications to nonlinearly perturbed queueing systems, population dynamics and epidemic models, and risk processes are considered. In this section, we discuss and comment on some new directions in the research concerned pseudo- and quasi-stationary phenomena for perturbed stochastic systems that relate to the theory developed in this book. We hope that this discussion will stimulate a further research in the area.

8.2.1 Nonlinearly perturbed stochastic systems with discrete time There is no doubt that all results on continuous time stochastic processes and systems presented in the book should have their analogues for discrete time stochastic processes and systems. It should be noted that discrete time models are interesting by themselves and have important applications. Some results concerned asymptotic expansions in mixed ergodic and large deviation theorems for nonlinearly perturbed regenerative processes with discrete time can be found in Englund and Silvestrov (1997) and Englund (2000, 2001), and in Silvestrov (2000a) for discrete time Markov chains with absorption. However, these results give discrete time analogues just for a small part of the results from the theory developed in the present book for continuous time processes. A development of a similar complete theory for nonlinearly perturbed stochastic processes and system with discrete time requires an additional comprehensive research.

8.2.2 Asymptotic expansions based on non-polynomial systems of infinitesimals Asymptotic expansions in mixed ergodic and limit/large deviation theorems and asymptotic expansions for quasi-stationary distributions can also be obtained for nonlinearly perturbed stochastic processes and systems where the expansions in the initial perturbation conditions are based on different systems of infinitesimals. In this book, all expansions are based on the integer powers 'n ."/ D "n , n D 0; 1; : : :, for the simplest infinitesimal '1 ."/ D ". This is a very natural system of infinitesimals that uses Taylor type expansions for nonlinearly perturbed characteristics of the corresponding processes and systems. A natural generalisation of the theory based on polynomial type infinitesimals can be developed for a modelQwhere the corresponding expansions include products of ni !i integer powers 'nN ."/ D m , nN D .n1 ; : : : ; nm /, n1 ; : : : ; nm D 0; 1; : : :, of iD1 "

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infinitesimals taken from some base set f 'i ."/ D "!i , i D 1; : : : ; m g, where !i , i D 1; : : : ; m, are positive real numbers. The difference in the corresponding expansions should mainly be caused by new forms in arithmetics for asymptotic expansions, i.e., in formulas for coefficients in sums, products, quotients, as well as in the power, exponential, and other functions in the asymptotic expansions. The importance of such expansions is that they permit to obtain a more dense net of infinitesimals in the corresponding expansions. We have a conjecture that analogous asymptotic expansions can also be constructed for a model Q where thencorresponding expansions include products of integer powers i N D .n ; : : : ; n /; n ; : : : ; n 'nN ."/ D m 1 m 1 m D 0; 1; : : :, m D 1; 2; : : :, of iD1 .'i ."// ; n infinitesimals taken from a finite or countable base set f 'i ."/; i D 1; 2; : : : g. The only condition should be imposed on these infinitesimals that any two infinitesimals of the product form given above, 'n0N 0 ."/ and 'n00N00 ."/, should be asymptotically comparable, i.e., there should be a constant 0  c  1 (determined by these infinitesimals) such that 'n0N 0 ."/='n00N0 ."/ ! c as " ! 0. An example of such expansions in mixed ergodic and large deviation theorems for regenerative processes and their applications to queueing systems can be found in Englund and Silvestrov (1997) and Englund (1999, 2000, 2001). In these works, nonlinear perturbations with the base set of the infinitesimals f"; e a=" g was considered.

8.2.3 Asymptotic expansions in mixed ergodic and large deviation theorems for semi-Markov type processes with countable and general phase spaces The asymptotic results obtained in Chapters 4 and 5 of this book relate to nonlinearly perturbed Markov chains and semi-Markov processes with finite phase spaces. These processes possess a regeneration property at return moments in a fixed state. This makes it possible to use the asymptotic results for nonlinearly perturbed regenerative processes presented in Chapter 3. Markov chains and semi-Markov processes with countable phase spaces possess similar regeneration properties. Moreover, the method of artificial regeneration developed in the works of Kovalenko (1977), Nummelin (1978a), Athreya and Ney (1978a), permits to construct regeneration moments for Markov and semi-Markov processes with a general phase space. In this way, the results concerning the asymptotic analysis of pseudo- and quasi-stationary phenomena for nonlinearly perturbed regenerative processes can be applied to nonlinearly perturbed Markov chains and semiMarkov processes with countable and general phase spaces. There exists a large number of works devoted to ergodic theorems, limit theorems for random functionals of hitting time types, as well as large deviation theorems for such functionals in non-mixed or mixed forms (with ergodic theorems). These works relate to Markov and semi-Markov type processes with finite, countable, and general

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phase spaces. The results presented in Chapter 4 are related to this direction. We list and make comments on the corresponding references in the bibliographical remarks. It should be noted that the results given in Chapter 4 play only a preparatory role with respect to the results related to exponential asymptotic expansions in mixed ergodic and large deviation theorem, and asymptotic expansions for quasi-stationary distributions given in Chapter 5. These theorems also require many auxiliary results that are important by themselves. Such results, for example, include asymptotic cyclic solidarity properties for the corresponding processes and asymptotic expansions for absorption and hitting probabilities, power and mixed power-exponential moments for the first hitting times and related functionals, and others. We believe that, altogether, these results make one of the main new authors’ contributions to the study of quasistationary phenomena for perturbed stochastic processes and systems. Most of these results have no analogues for Markov and semi-Markov type processes with countable and general phase spaces, and constitute a new direction for the future research. As usual, operator analogues for matrix techniques used in the models with finite phase spaces should be employed for models with countable and general phase spaces, which significantly complicates the problem under consideration.

8.2.4 Mixed ergodic and large deviation theorems for semi-Markov type processes with asymptotically uncoupled phase space Results presented in Chapters 4 and 5 are related to a basic model of nonlinearly perturbed semi-Markov processes with one limit class of recurrent-without-absorption states. It would be very interesting to extend the theory to models in which the set of non-absorption states for perturbed processes asymptotically splits into several classes of recurrent-without-absorption states which do not communicate with each others. We give corresponding references in the bibliographical remarks. It should be noted that most of the results for models with an asymptotically uncoupled phase space are related to weak convergence limit theorems for distributions of hitting times for pseudo-stationary models mixed with the corresponding ergodic theorems and related theorems on asymptotic aggregation of Markov and semi-Markov type processes. Also a few results concerned asymptotic expansions for linearly perturbed Markov and semi-Markov type processes are known. At the same time, a study of large deviation theorems for hitting times, as well as exponential asymptotic expansions in mixed ergodic and large deviation theorems for hitting times and asymptotic expansions for quasi-stationary distributions in models with asymptotically uncoupled sets of recurrent-without-absorption states for linearly and all the more nonlinearly perturbed and Markov and semi-Markov type processes, have not been conducted before. Such theorems certainly constitute an additional prospective direction for the future research. In principle, the method based on applying asymptotic results to perturbed regenerative processes developed in Chapter 3 can be used in combination with the method

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of recurrence asymptotic analysis for power and mixed power-exponential moments of hitting times developed in Chapter 5. However, one should expect that pseudoabsorption effects caused by asymptotic vanishing of communication between different recurrent-without-absorption classes of states may complicate the asymptotic analysis for power and mixed power-exponential moments of hitting times for models with asymptotically uncoupled phase space.

8.2.5 Mixed ergodic and large deviation theorems for stochastic processes with semi-Markov modulation (switchings) Methods of the asymptotic analysis of pseudo- and quasi-stationary phenomena in nonlinearly perturbed Markov chains and semi-Markov processes also can be applied to more general processes of Markov and semi-Markov types. In order to be able to apply the methods of asymptotic analysis developed for perturbed renewal equations, such processes should possess appropriate regeneration properties. In particular, these are so-called stochastic processes with semi-Markov modulation (switchings), which possess imbedded semi-Markov processes. In the Markov case, these processes are also known as Markov processes with discrete components. We refer to the book of Gikhman and Skorokhod (1975), Silvestrov (1980a), Koroliuk and Limnios (2005), and Anisimov (2008), where one can find descriptions of these classes of stochastic processes. Sections 6.2 and 6.3 contain a detailed description of a typical example of M/G queueing systems with quick service and a bounded queue buffer, which can be described with the use of processes with semi-Markov modulation. This example shows in which way the methods developed in the book can be applied to the asymptotic analysis of pseudo- and quasi-stationary phenomena in nonlinearly perturbed processes with semi-Markov modulation. The main point in the corresponding asymptotic analysis is to express the perturbation and other conditions in terms of local characteristics related to the individual semi-Markov regeneration cycles. We have no doubts that the main results of the theory presented in this book can be extended to this class of processes that are more general than regenerative and semiMarkov processes.

8.2.6 Double asymptotic expansions in limit and large deviation theorems for lifetimes in nonlinearly perturbed stochastic processes and systems The expansions obtained in Chapter 5, applied to marginal distributions of absorption r r times, have the form Pf"."/ > t ="r 1 g=e .0 Ca1 "CCar " /t=" D 1 C o.1/ for r D 1; : : : ; k. These expansions describe the behaviour of relative errors in large deviation zones of different orders, when approximating the distribution of absorption times with exponential type distributions that have specially fitted parameters.

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Our conjecture is that it should also be possible to expand the residual term o.1/ in the asymptotic relation given above. Our hope is based on the results given in Korolyuk, Penev and Turbin, (1972), Poliˇscˇ uk and Turbin (1973), and Silvestrov and Abadov (1991, 1993), where such expansions have been obtained in pseudo-stationary model (0 D 0), respectively, for the case r D 1 in the first two papers and for the cases r D 1 and r D 2 in the last two papers. Note that the case r D 1 corresponds to asymptotic expansions in usual weak limit theorem while the case r D 2 corresponds asymptotic expansions large deviation theorems for the zone of large deviation of the order O."1 /. In the cases r > 2, the question about such “double” expansions is an open problem. These remarks can be also related to exponential expansions in mixed ergodic and large deviation theorems.

8.2.7 Explicit upper bounds for the remainder terms in asymptotic expansions for nonlinearly perturbed stochastic systems The problem of finding explicit upper bounds for remainder terms in exponential asymptotic expansions in mixed ergodic and limit/large deviation theorems for nonlinearly perturbed stochastic processes and systems is of a special importance. For example, such explicit upper bounds for the remainder terms are obtained in the asymptotic expansions given in the papers Silvestrov and Abadov (1991, 1993) mentioned above. Of course, the research of the rates of convergence in limit theorems for lifetime functionals for regenerative processes and Markov- and semi-Markov type processes (which correspond to the problem of finding explicit upper bounds for the remainder terms in zero-order expansions) continue to be actual. The corresponding references related to these problems are given in the bibliographical remarks. Explicit upper bounds for the remainder terms in asymptotic expansions of stationary and quasi-stationary distributions are known for linearly perturbed Markovand semi-Markov processes. Similar problems related to explicit upper bounds for the remainder terms in asymptotic expansions of stationary and quasi-stationary distributions for nonlinearly perturbed stochastic processes and systems have not been conducted so far.

8.2.8 Quasi-stationary phenomena in nonlinearly perturbed queueing and reliability systems Asymptotic analysis of pseudo- and quasi-stationary phenomena in nonlinearly perturbed queueing systems and networks of M/M, M/G, and G/M types, which can be described with the use of Markov chains, semi-Markov processes, and stochastic processes with semi-Markov modulation, is an unlimited area for applications of the theory developed in the book. Examples of such applications are given in Sections 6.1– 6.4. These examples show that, in such applications, the main problems translate to technical and sometimes non-trivial and interesting calculations connected with a reformulation of the perturbation conditions and rewriting formulas for the coefficients

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of the asymptotic expansions and quasi-stationary distributions in terms of local characteristics used to define the corresponding queueing systems. It would be interesting to develop an asymptotic analysis of pseudo- and quasistationary phenomena for nonlinearly perturbed queueing systems with a more general structure of input flows of customers and service processes, e.g., systems with input flows of group income of customers and group service, different models of vacations, systems with parameters in input flows and service processes depending on a queue, systems with modulated input flows and switching service regimes, systems with several kinds of customers, systems with several types of servers, with and without reservation, etc. As well known, queueing systems of G/G type can also be described with the use of regenerative processes. Here the moments when the queue becomes empty play, as a rule, the role of regeneration times. Thus, we can use the methods of asymptotic analysis of pseudo- and quasi-stationary phenomena for perturbed regenerative processes. However, the analysis of regeneration cycles and a reduction of the conditions and formulas that are based on the global characteristics connected with the regeneration cycles to, correspondingly, conditions and formulas that would be based on local characteristics used to define the corresponding systems is expected to be much more difficult for such systems.

8.2.9 Quasi-stationary phenomena in nonlinearly perturbed models of biological type systems Another unlimited area for applications of the asymptotic results dealing with pseudoand quasi-stationary phenomena in stochastic systems are nonlinearly perturbed epidemic and population dynamics models which can be described with the use of Markov chains, semi-Markov processes with finite phase spaces. Examples of such applications are given in Sections 6.4 and 6.5. Similarly to the examples from the queueing theory, the main problems are to reformulate the perturbation conditions and to express the formulas for the coefficients in the asymptotic expansions and quasi-stationary distributions in terms of the local characteristic used to define the corresponding epidemic or population dynamics models. Nonlinearly perturbed epidemic and population dynamics models with different types, e.g., sex, age, health status, etc., of individuals in the population, forms of interactions between individuals, environmental modulation, complex spatial structure, etc., make interesting objects for asymptotic analysis of pseudo- and quasi-stationary phenomena.

8.2.10 Asymptotics of probabilities and other characteristics connected with ruins for perturbed risk type processes Chapter 7 of the book is devoted to studies of quasi-stationary phenomena for nonlinearly perturbed risk processes. Asymptotic results for ruin probabilities, densities of

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non-ruin probabilities, and distributions of the capital surplus prior to and at the time of a ruin are given. However, there are many other functional characteristics connected with risk processes that satisfy renewal equations. In such cases, analogous asymptotic results can be obtained with the use of asymptotic methods developed for perturbed renewal equations in Chapters 1 and 2. More general risk type processes may also supply models with functional characteristics satisfying renewal equations. For example, such are risk processes with investment components described by Brownian motions, more general L´evy type risk processes, risk processes with modulated claim flows, etc.

8.2.11 Markov and semi-Markov processes describing dynamics of credit ratings Finally, we would like also to mention a possible prospective area of financial applications for asymptotic methods developed in the book. The methods of asymptotic analysis of pseudo- and quasi-stationary characteristics may be also applied to studies of default type distributions for Markov and semi-Markov processes with absorption which become common to use for description of dynamics of credit ratings.

8.2.12 Numerical studies connected with asymptotic expansions describing quasi-stationary phenomena in nonlinearly perturbed stochastic systems Numerical and simulation studies make a natural supplement to analytical results, especially such as asymptotic expansions for functional characteristics of perturbed stochastic processes and systems. For example, asymptotic results presented in Section 7.4 yield formulas for approximations of ruin probabilities based on higher order moments of the claim distributions. These approximations are asymptotically optimal in the sense that the corresponding relative errors tend to zero. It would be interesting to compare them with other known approximations. This, however, would require to carry out additional comprehensive analytical, numerical, and simulation studies that are beyond the frame of the present book. We hope that these experimental studies will be realised in a future for this model as well as for other perturbed models of stochastic processes and systems.

8.2.13 Asymptotics for perturbed equations of renewal type The program for studies of pseudo- and stationary phenomena in nonlinearly perturbed stochastic processes and systems realised in the book is based on the method of the asymptotic analysis for perturbed renewal equations presented in Chapters 1 and 2. We have no doubts that the main asymptotic results for the perturbed renewal equations presented in these chapters should hold for more general equations of renewal type. In many cases, such results can be achieved by a suitable reduction to the case

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of classical renewal equation treated in the book. For example, Shurenkov (1980a, 1980b, 1980c) gave such a generalisation of some results presented in Chapter 1 to the case of a matrix renewal equation. Chapter 4 contains a more through and extended presentation of matrix version of asymptotic results for perturbed renewal equation given in Chapter 1 for the model case of perturbed semi-Markov processes with absorption. Chapter 5 essentially improves these asymptotic results to the much more advanced form of the corresponding asymptotic expansions. In fact, this chapter contains a matrix generalisation of the results of Chapter 2 to the model case of nonlinearly perturbed semi-Markov processes with absorption. There exist other versions of renewal type equations, for example, the so-called renewal equations with waits, operator renewal type equations, renewal equations on a whole line etc. It would be interesting to extend the theory presented in the present book to such equations. There are also other areas, beyond ergodic, pseudo- and quasi-stationary problems, where the renewal equation and the corresponding asymptotic results play an essential role. For example, these are asymptotic problems for moment functionals appearing in the theory of branching processes and many others. We hope that the book will stimulate future works directed to new applications of the theory presented in this book.

8.3 Bibliographical remarks Our bibliographical remarks are structured by chapters according to the contents of the book. We first refer to works that contain results given in the corresponding chapters and indicate new results included in the book and published for the first time. Then we refer to some works that contain results directly connected with the results given in the corresponding chapters. We also collect and give references to works in related areas, dealing with ergodic and quasi-ergodic theorems, stability theorems, limit and large deviation theorems for lifetime-type functionals and asymptotic aggregation theorems for regenerative, Markov, and semi-Markov type processes, as well as applications of such theorems to queueing systems, models of population dynamics, epidemic models, and other stochastic systems. Chapters 1. Chapter 1 presents results related to a generalisation of the classical renewal theory to a model of the perturbed renewal equation. We begin with the references concerning the classical renewal theorem. The final form of this important theorem (see, Theorem 1.1.1), describing the asymptotics of a solution of the renewal equation, was given in Feller (1966). Chapter 1 is based on the results obtained in Silvestrov (1976, 1978, 1979a), where a generalisation of the renewal theorem to a model of the perturbed renewal equation was given. The main results contained in these papers are given in Theorems 1.2.1 and 1.3.1, which are direct analogues of the classical renewal theorems for a model of a perturbed renewal equation, and also in Theorems 1.4.1 and 1.4.2, which describe the

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asymptotics of a solution of a perturbed renewal equation under the Cram´er type moment conditions imposed on the distributions that generate the corresponding renewal equations. Shurenkov (1980a, 1980b, 1980c) has extended some of the results mentioned above to a model of perturbed matrix renewal equations. Englund and Silvestrov (1997) and Englund (2000, 2001) gave variants of the corresponding results for a discrete time perturbed renewal equation. References to some selected works related to asymptotic renewal theory are Feller (1941, 1949, 1950, 1961, 1966), T¨acklind (1945), Blackwell (1948, 1953), Erd¨os, Feller and Pollard (1949), Cox and Smith (1953), Smith (1954, 1955b, 1958, 1961a, 1962), Dynkin (1955), Feller and Orey (1961), Farrell (1962), Garsia and Lamperti (1962/1963), Garsia (1963), Borovkov (1964), Stone (1965a, 1965b, 1966), Heyde (1967b), Stone and Wainger (1967), Teugels (1967, 1968a, 1970, 1990), Rogozin (1973, 1976), Sevast’yanov (1974), Zubkov (1975), Silvestrov (1976, 1978, 1979a, 1980c, 1980e, 1983a, 1984a, 1984b, 1994), Arjas, Nummelin and Tweedie (1978), Kalasnikov (1978b), Kartashov (1978, 1979a, 1982a, 1982b), Silvestrov and Banakh (1981), Thorisson (1987), Bertoin (1999), Grey (2001), and Yin Chuancun and Zhao Junsheng (2006). We would also like to refer to works, where asymptotic results in the renewal theory were extended to more general renewal type models and equations, Karlin (1955), Smith (1955a, 1960, 1961b, 1969), Hatori (1959, 1960), Farrell (1964, 1966a, 1966b), Gerber (1970), Su Yeong Tzay (1993), Woodroofe (1982), Gut (1983, 1988), Janson (1983a), Alsmeyer (1986, 1987, 1988a, 1988b, 1988c, 1991a, 1991b, 1992, 1994a, 1994b, 1995, 1996), Alsmeyer and Irle (1986), Lam and Lehoczky (1991), Schmidli (1997), Su Yeong-Tzay (1998), Su Yeong Tzay and Wang Chi Tshung (1998), Kim Dong Yun and Woodroofe (2003), Sgibnev (2003, 2006), and Kartashov and Stroev (2005). Results presented in Chapter 1 form a basis for studying a nonlinearly perturbed renewal equation, which is done in Chapter 2. Chapters 2. In this chapter, we present exponential expansions for solutions of nonlinearly perturbed renewal equations. These expansions essentially improve the asymptotics of solutions of perturbed renewal equations given in Chapter 1. Chapter 2 is based on the results obtained in the papers Silvestrov (1995) and Gyllenberg and Silvestrov (1998, 1999a, 2000a), and on extensions of these results to a model of a perturbed renewal equation satisfying more general perturbation conditions. In Silvestrov (1995), nonlinear perturbation conditions were imposed on the moments of the distribution functions that generate the corresponding renewal equation. Explicit expansions for the corresponding characteristic roots were given, and the corresponding exponential expansions were obtained for solutions of nonlinearly perturbed renewal equations. These results are given in Theorems 2.1.1 and 2.1.2. Theorems mentioned above cover the case of an asymptotically proper renewal equation. The general case, where the limit renewal equation can be proper or improper,

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was considered in Gyllenberg and Silvestrov (1998, 1999a, 2000a). The corresponding result is formulated in Theorem 2.2.1. The latter paper also contains asymptotical expansions for the renewal limit given in Theorem 2.3.2. Theorems 2.1.3 and 2.2.2 present new results that generalise Theorems 2.1.1, 2.1.2, and 2.2.1 mentioned above to models with more general perturbation conditions. The results mentioned above are related to a model of a continuous time renewal equation. The case of a discrete time renewal equation is interesting on its own right. Some results are given in the papers Englund and Silvestrov (1997) and Englund (2000, 2001). Also, exponential expansions, which are based on non-polynomial perturbation conditions, were considered for discrete and continuous time perturbed renewal equations in Englund and Silvestrov (1997), and Englund (1999a, 1999b, 2001). The results presented in Chapters 1 and 2 make a basis for our studies of pseudoand quasi-stationary phenomena in nonlinearly perturbed regenerative processes. Chapter 3. In Chapter 3, we present applications of the asymptotic results obtained in Chapters 1 and 2 to a model of nonlinearly perturbed regenerative processes. Chapter 3 is based on the results obtained in the papers Silvestrov (1995) and Gyllenberg, and Silvestrov (1998, 1999a, 2000a), as well as on extensions of these results to a model of perturbed regenerative processes with more general perturbation conditions and regenerative processes with transition period. We formulate mixed ergodic and limit/large deviation theorems for joint distributions of positions of perturbed regenerative processes and regeneration stopping times. This is done in Theorems 3.2.1–3.3.6, which describe pseudo-stationary and quasistationary phenomena for perturbed regenerative processes. In fact, these theorems are direct corollaries of the renewal theorem for perturbed renewal equations proved in Silvestrov (1976, 1978, 1979a) and given in Chapter 1. These results have analogues in the literature. In particular, some slight modifications were given in the papers Shurenkov (1980a, 1980b, 1980c). Some similar results related for discrete time nonlinearly perturbed regenerative processes as well as some results related to mixed ergodic and limit/large deviation theorems for nonlinearly perturbed regenerative processes with non-polynomial perturbations can be found in Englund and Silvestrov (1997) and Englund (2000, 2001). Our main new achievements are Theorems 3.4.2 and 3.4.4 that are based on the results of the papers Silvestrov (1995) and Gyllenberg and Silvestrov (2000a). The main new elements are nonlinear perturbations and higher order exponential expansions, which cover large deviation theorems for regenerative stopping times in various large deviation zones that are determined from the balancing conditions imposed on the rate of growth of time and the rate of perturbation. Also, a new element is a mixed ergodic and large deviation form of the corresponding theorems that describe asymptotics for joint distribution of the position of regenerative process and the regeneration stopping time. Theorems 3.4.3 and 3.4.5 present new results that generalise Theorems 3.4.2 and 3.4.4 mentioned above, covering the cases with more general perturbation conditions, while Theorems 3.4.6–3.4.10 extend all the above theorems to a model of

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nonlinearly perturbed regenerative processes with transition period. Theorems 3.5.1, 3.5.2, and 3.5.4 also contain new results on asymptotic expansions for quasi-stationary distributions of nonlinearly perturbed regenerative processes. We mentioned above some papers which contain results related to mixed ergodic and limit/large deviation theorems for regenerative processes. These theorems describe asymptotics of the joint distribution of the position of regenerative process and the regeneration stopping time. In a marginal form, these theorems reduce either to individual ergodic theorems for perturbed regenerative processes, which describe asymptotics of distributions of regenerative processes, or to limit and large deviation theorems for regeneration stopping times. Let us mention some works on ergodic theorems for regenerative processes as well as rates of convergence in these theorems and stability of stationary characteristics. The stochastic process setting and use of some constructive probabilistic methods such as coupling slightly distingue these results of the corresponding asymptotic results in renewal theory. These works are Smith (1955a, 1958), Lamperti (1961, 1962), Feller (1966), Lindvall (1977, 1979a, 1982, 1983, 1986, 1992a, 1992b), Silvestrov (1976, 1978, 1979a, 1980c, 1980e, 1983a, 1984a, 1984b, 1994), Kalashnikov (1978b, 1981a, 1990, 1994b, 1994c), Banakh (1981, 1982a, 1982b), Thorisson (1983, 1985a, 1992, 1998, 2000), Anichkin (1985), Lindvall and Rogers (1996), Silvestrov and Stenflo (1998), Stenflo (1998), and Glynn and Thorisson (2004). A regeneration stopping time can be represented in a form of a geometric random sum with a reminder term, as was shown in Section 3.1. That is why, limit theorems for regeneration stopping times are closely connected with limit theorems for geometric random sums that were a subject of thorough studies in R´enyi (1956), Kovalenko (1965), Keilson (1966a, 1966b, 1978), Gnedenko and Kovalenko (1964), Gnedenko and Frayer (1969), Silvestrov (1969, 1980a), Masol (1973, 1974, 1975a, 1975b, 1976a, 1977), Solov’ev (1971), Korolyuk, D. (1982), Kruglov and Korolev (1990), Klebanov, Melamed, and Umarov (1991), Gnedenko and Korolev (1996), Silvestrov (2004a), Silvestrov and Teugels (2004), and Silvestrov and Drozdenko (2005, 2006). We refer to the book Silvestrov (2004a) for a detailed list of references. The rates of convergence and large deviation theorems in limit theorems for regeneration stopping times and geometric sums and related results were studied in Feller (1966), Silvestrov (1969a, 1969c, 1980a, 1995), Freyer (1970), Solov’ev (1971), Szynal (1976), Kovalenko (1980, 1994), Kalashnikov (1983, 1986, 1993, 1994b, 1997), Khusanbayev (1983, 1984), Kalashnikov and Vsekhsvyatski (1985, 1989), Vsekhsvyatski and Kalashnikov (1985a), Grishechkin (1986), Korolev (1987, 1989a, 1989b), Melamed (1987, 1989), Kovalenko and Kuznetsov (1988), Brown (1990), Hsu Guanghui and He Qiming (1990), Kruglov and Korolev (1990), Sudakova (1990, 1993), Kartashov (1991, 1996b), Willekens and Teugels (1992), Gnedenko and Korolev (1996), Kalashnikov and Konstantinidis (1996), Asmussen and Kalashnikov (1999), Bon and Kalashnikov (1999, 2001), Kalashnikov and Tsitsiashvili (1999a, 1999b, 2001), Ben-

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ing and Korolev (2002), and Glynn and Thorisson (2004). It should be noted that the works listed above give the usual asymptotics in weak limit theorems and the corresponding estimates for the rate of convergence in the pseudo-stationary case, where the stopping probabilities for one regeneration cycle (or success probability for the geometric random index) tend to zero, while results concerning large deviation asymptotics mainly relate to the quasi-stationary case, where the stopping probabilities for one regeneration cycle (or success probability for geometric the random index) are separated from zero. Both the models with light and heavy tail distributions of summands were considered. The results on weak convergence and large deviation asymptotics for regeneration stopping times given in Theorems 3.2.1–3.3.6 deal with the case of asymptotically ergodic regenerative processes with light tail distributions of regeneration moments, but give these results in a united form covering both the pseudo- and quasi-stationary cases. The exponential expansions in the mixed ergodic and limit or large deviation theorems for absorption times and the asymptotic expansions of the quasi-stationary distributions for nonlinearly perturbed regenerative processes, which are given in Theorems 3.4.2–3.5.4, have no analogues in the works listed above. We would like also to mention some basic papers and books on renewal theory and regenerative processes and related problems that are Smith (1955b), Cox (1962), Kingman (1972), Silvestrov (1980a), Shedler (1987, 1993), Gut (1988), Bingham, Goldie, and Teugels (1989), Shurenkov (1989), Alsmeyer, (1991b), Kalashnikov (1994b, 1997), and Thorisson (2000). The results presented in Chapter 3 make a basis for our studies of pseudo- and quasi-stationary phenomena in nonlinearly perturbed semi-Markov processes. Chapter 4. We make here a very detailed analysis of pseudo- and quasi-stationary phenomena for perturbed semi-Markov processes with a finite set of states and absorption. Results presented in Chapter 4 extend the corresponding results obtained in Gyllenberg and Silvestrov (1999a, 2000a), and Silvestrov (2000a). There are two classes of perturbed models that differ in the asymptotical properties of the absorption probabilities. The first class includes models with small local absorption probabilities. More precisely, these probabilities are functions of a perturbation parameter " and tend to zero if " tends to zero. In such models, the absorption times ."/ tend in probability to infinity as " ! 0. The second class includes models with some of the local absorption probabilities separated from zero. In such models, the absorption times ."/ are asymptotically stochastically bounded random variables as " ! 0. To distinguish these two cases, we introduce the term pseudo-stationary for mixed limit/large deviation theorems in the first case, while the term quasi-stationary is traditionally used in the second case. In Chapter 4 we consider mixed ergodic and limit theorems and mixed ergodic and large deviation theorems for perturbed semi-Markov processes in the usual triangular array models, where the perturbation conditions are given in the simplest form that

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assumes convergence of local transition probabilities of the perturbed processes to the corresponding characteristics of the limit processes as the perturbation parameter " ! 0. The main results are Theorems 4.4.1 and 4.4.2, which represent mixed ergodic and limit theorems for perturbed semi-Markov processes under minimal moment conditions on the sojourn times, while Theorems 4.6.1–4.6.2 and 4.6.3–4.6.4, which represent mixed ergodic and large deviation theorems for perturbed semi-Markov processes under Cram´er type moment conditions imposed on the sojourn times, respectively, for pseudo-stationary and quasi-stationary models. Also, Theorems 4.6.5 and 4.6.9 give conditions for convergence of conditional distributions of the processes, assuming a non-absorption condition, to the corresponding quasi-stationary distributions, while Theorems 4.6.6 and 4.6.10 give conditions for convergence of quasi-stationary distributions for perturbed semi-Markov processes. Results given in Shurenkov (1980a, 1980b, 1980c), and Alimov and Shurenkov (1990a, 1990b), Shurenkov and Degtyar (1994), and Ele˘ıko and Shurenkov (1995b) are the most close with the results presented in Chapter 4. In the earlier works these authors used the method based on the use of regeneration properties of semi-Markov processes and reduction of the analysis to the case of scalar perturbed renewal equation, then switched to methods that are based on a direct asymptotic analysis of matrix or general Markov perturbed renewal equations. We do prefer to use the former regeneration based method. This method has a significant advantage, since it allows to not only get the corresponding mixed ergodic and limit/large deviation theorems but also improve them to a form of exponential asymptotic expansions as well as get asymptotic expansions for quasi-stationary distributions. Such expansions are hardly expected to be obtained with a use of direct matrix calculations that takes an awkward form. We would like also to refer to works Silvestrov (1981), Abadov (1984), and Silvestrov and Abadov (1984, 1991, 1993), where large deviation theorems and the corresponding asymptotic expansions for distributions of hitting times for perturbed Markov chains were given under pseudo-stationary conditions. These results stimulated ours studies related to exponential asymptotics in large deviation theorems for perturbed semi-Markov processes. There exist a huge number of works in related areas. We first would like to refer to papers that are related to pseudo-stationary asymptotics. The model with finite phase space was most extensively investigated. Limit theorems for random functionals, known as first hitting times, first passage times, first record times, etc., and closely related to problems of asymptotic aggregation for Markov type processes, have been investigated by many researchers. Together with the limit theorems, the corresponding estimates for the rate of convergence in the models with coupled and nearly uncoupled phase space have been intensively studied. The characteristic feature of these results is that some explicit algebraic algorithms can be given for calculating the parameters of limit law, constants in the corresponding estimates, etc. Limit theorems, rates of convergence for distributions of hitting times and related problems were studied in Harris (1952), Meshalkin (1958), Keilson (1966b, 1974,

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1978, 1998), Solov’ev (1966), Darroch and Morris (1967), Korolyuk (1969), Silvestrov (1969c, 1970, 1971a, 1971b, 1972c, 1974, 1980a), Anisimov (1970, 1971a, 1971b), Korolyuk and Turbin (1970, 1976a), Holst (1971, 1977, 1979, 2001), Turbin (1971, 1972), Masol and Silvestrov (1972), Zakusilo (1972a, 1972b), Kovalenko (1973, 1975), Zubkov, and Miha˘ılov (1974), Masol (1976b), Brown (1983), Brown and Guang Ping (1984), Gaver, Jacobs, and Latouche (1984), Gut and Holst (1984), Latouche, Jacobs, and Gaver (1984), Aldous (1986), Brown and Shao (1987), Korolyuk, D. (1990), Latouche (1991), Asmussen (1994), Gyllenberg and Silvestrov (1994, 1999a, 2000a), Chen Dayue, Feng Jianfeng, and Qian Minping (1996), Stewart (1997, 1998, 2001, 2003), Ferrari and Garcia (1998), Keilson and Vasicek (1998), Yin and Zhang (1998), and Zhang and Yin (2004), and Silvestrov and Drozdenko (2005, 2006). Related problems of asymptotic aggregation for Markov chains and semi-Markov processes with finite phase space were studied in Meshalkin (1958), Simon and Ando (1961), Hanen (1963a, 1963b, 1963c, 1963d), Anisimov (1972, 1973), Courtois (1975, 1977), Gaitsgory and Pervozvansky (1975), Kalashnikov (1978a), Anisimov and Sityuk (1980), Delebecque and Quadrat (1981a, 1981b), Coderch, Willsky, Sastry, and Castanon (1983a, 1983b), Delebecque (1983a, 1983b), Stewart (1983, 1984, 1991, 1993, 1998, 2001), Delebecque, Quadrat, and Kokotovi´c (1984), Schweitzer (1984, 1991), Cao and Stewart (1985), Schweitzer, Puterman, and Kindle (1985), Schweitzer and Kindle (1986a, 1986b), Rohlicek (1987), Rohlicek and Willsky (1988a, 1988b), Stewart and Sun (1990), Kim and Smith (1991), Stewart and Zhang (1991a, 1991b), Sumita and Rieders (1991), Ledoux, Rubino, and Sericola (1994), Stewart, G., Stewart, W., and McAllister (1994), Stewart (1998, 2001), Yin and Zhang (1998), Yin, Zhang, and Badowski (2000a, 2000b, 2000c), Badowski, Yin, and Zhang (2003), and Yin and Zhang (2003). Similar problems for Markov chains and Markov type processes with countable and general phase space, in particular, limit theorems for hitting times and rates of convergence were obtained in Gaver (1964, 1976), Heyde (1964, 1966, 1967a), Silvestrov (1969b, 1969c, 1972a, 1972b, 1972c, 1974, 1979b, 1980a, 1981, 1995, 2000a), Ros´en (1970, 1978), Gani and Lehoczky (1971), Basu (1972, 1978), Gani (1972), Solov’ev (1972), Chow and Hsiung (1976), Ahmad, and Basu (1983), Gut (1974a, 1974b, 1975, 1983, 1988, 1990a, 1990b), Mileshina (1975, 1977), Korolyuk and Turbin (1976b, 1978), Kalashnikov (1978b, 1986, 1994a, 1994b), Vo˘ına (1978), Kaplan (1979a, 1979b, 1980), Korolyuk, Turbin, and Tomusjak (1979), Zubkov (1979), Chow, Hsiung, and Yu, (1980), Wang Zi Kun (1980), Aldous (1982, 1989, 1991), Janson (1983b), Korolyuk, D. and Silvestrov (1983, 1984), Vsekhsvyatski and Kalashnikov (1985b), Lalley (1984), Kartashov (1986, 1987, 1996b), Martin-L¨of (1986a), Anisimov (1988), Borovkov (1998b, 1998c), Silvestrov and Velikii (1988), Kostava and Shurenkov (1988), Todorovic and Gani (1989), Korolyuk, D. (1990), Aldous and Brown (1992, 1993), Ferrari, Galves, and Landim (1994), Ferrari, Galves, and Liggett (1995), Sevast’yanov (1995, 1998, 2000, 2006), Motsa and Silvestrov (1996),

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Walker (1998), Gasanenko (1999, 2001), Kesten and Maller (1999), Korolyuk (1999), Korolyuk and Limnios (2005), Puchała and Rolski (2005), and Asselah and Ferrari (2006). The asymptotic aggregation and related problems were studied in Gusak and Korolyuk (1971), Anisimov (1975, 1980, 1984, 1986, 1988, 2008), Korolyuk and Turbin (1976a, 1978, 1982), Vo˘ına (1978, 1979), Arias and Korolyuk (1980), Kalashnikov (1981a), Anisimov, Vo˘ına, and Lebedev (1983), Bobrova (1983), Korolyuk, V.V. (1983), Korolyuk, V.S and Korolyuk, V.V. (1999), Kartashov (1986, 1987, 1996b), Korolyuk (1986, 1993, 1997), Korolyuk and Limnios (1998, 1999, 2000, 2002, 2004a, 2004b, 2005), Korolyuk and Svishchuk (1990, 1992, 1995), Al-Eideh (1995), Kovalenko, Birolini, Kuznetsov, and Shenbukher (1998), and Korolyuk and Mamonova (2003). Ergodic theorems and the corresponding rates of convergence were obtained for discrete and continuous time Markov chains in Markov (1906), Kolmogorov (1931, 1937), Doeblin (1936, 1937, 1938, 1940), L´evy (1951), Harris (1956), Kendall (1959), Kingman (1963), Vere-Jones (1962, 1963, 1964, 1967, 1968), Kesten and Spitzer (1963), Kesten (1963, 1970), Cheong (1967), Teugels (1968c, 1972), Orey (1971), Callaert and Keilson (1973), Pitman (1974), Griffeath (1974/75, 1976, 1978), Revuz (1975), Nummelin and Tweedie (1976), Kovalenko (1977), Athreya and Ney (1978b), Keilson (1978, 1998), Nummelin (1978a, 1984), Seneta (1979, 1984), Borovkov (1980, 1998a), Brockwell, Gani, and Resnick (1982, 1983), Nummelin and Tuominen (1982), Tan Choon Peng (1982, 1983), Tweedie (1983a, 1983b, 2001), Kartashov (1984, 1985d, 1996b, 1996c), Gani and Tin Pyke (1985), van Doorn (1985, 2001, 2003), Chen (1991, 1996, 2005), Semal (1991), Ze˘ıfman (1991b), Borovkov and Foss (1992, 1994), Lindvall (1992a), Meyn and Tweedie (1992, 1993a, 1993b, 1993c, 1994a, 1994b), Spieksma and Tweedie (1994), Tuominen and Tweedie (1994), Down, Meyn, and Tweedie (1995), Kesten (1995), Rosenthal (1995), Ze˘ıfman (1995), Hoppensteadt, Salehi, and Skorokhod (1996a, 1996b), Lund and Tweedie (1996), Lund, Meyn, and Tweedie (1996), Mengersen and Tweedie (1996), Roberts and Tweedie (1996, 1999, 2001), Scott and Tweedie (1996), Granovski˘ı and Ze˘ıfman (1997, 2000, 2004), Roberts and Rosenthal (1997), Keilson and Vasicek (1998), Corcoran and Tweedie (2001), Guglielmi and Tweedie (2001), Jarner and Tweedie (2001, 2002, 2003), Zhang, Chen, Lin and Hou (2001), Fort and Moulines (2003), Borovkov and Hordijk (2004), Elesin, Kuznetsov, and Ze˘ıfman (2004), Andrieu and Moulines (2006), Mao (2006), Roberts, Rosenthal, Segers, and Sousa (2006), Ze˘ıfman, Leorato, Orsingher, Satin, and Shilova (2006), Roberts and Rosenthal (2007), and Sirl, Zhang Hanjun, and Pollett (2007). Similar problems for more general semi-Markov and related processes were studied in Nagaev (1965, 1982), Teugels (1968b, 1970), C¸inlar (1969a, 1969b, 1974a, 1974b, 1975), Shur (1967, 1976, 1977, 1984, 1985, 2002, 2005), Keilson (1969), Jacod (1970, 1971), Cheong and Teugels (1973), Kesten (1974), Gikhman and Skorokhod (1975), Ezhov and Shurenkov (1976), Athreya and Ney (1978b), Athreya, McDonald, and

498

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Supplements

Ney (1978), Nummelin (1978b), Ele˘ıko (1980, 1990, 1998), Silvestrov (1980a, 1980b, 1983c, 1994), Shurenkov (1981, 1983, 1989), Silvestrov and Pezhinska (1984), Kartashov (1985c, 1996a, 1997, 2000, 2005), Anderson and Athreya (1987), Aldous (1988), Ethier and Kurtz (1988), Meyn (1989), Alimov and Shurenkov (1990b), Meyn and Caines (1990), Thorisson (1990, 1998, 2000), Asmussen (1992), Meyn and Guo Lei (1993), Alimov (1994a, 1994b), Alsmeyer (1994c, 1997), Ele˘ıko and Shurenkov (1995a), Sgibnev (1996), Stenflo (1996, 1998), Hoppensteadt, Salehi, and Skorokhod (1997), Iosifescu (1999), Konstantopoulos and Last (1999), Balaji and Meyn (2000), Roberts and Tweedie (2000), Alsmeyer and Hoefs (2001, 2002, 2004), Fuh Cheng-Der and Lai Tze Leung (2001), Kl¨uppelberg and Pergamenchtchikov (2003), Kontoyiannis and Meyn (2003, 2005), Fuh Cheng-Der (2004), and Baxendale (2005). Stability problems, rates of approximation for the perturbation of stationary distributions and related characteristics for Markov chains and semi-Markov type processes were studied in Seneta (1967, 1968, 1973, 1980, 1981, 1988, 1991, 1993a, 1991, 1993b, 1993c), Schweitzer (1968), Kurtz (1971), Tweedie (1971, 1975a, 1975b, 1980), Golub and Seneta (1974), Courtois (1975, 1977), Courtois and Louchard (1976), Allen, Anderssen, and Seneta (1977), Borovkov (1977), Kalashnikov (1978b, 1994a), Kalashnikov and Anichkin (1981), Aissani and Kartashov (1983b), Cocozza-Thivent, Kipnis, and Roussignol (1983), Courtois and Semal (1984a, 1984b, 1984c 1991), Haviv and Rothblum (1984), Haviv and Van der Heyden (1984), Kartashov (1985b, 1996b), Hunter (1986), Haviv (1986, 1988, 1992, 1999), Haviv and Ritov (1986, 1993), Burnley (1987), Gibson and Seneta (1987), Johnson and Isaacson (1988), Ze˘ıfman (1989a, 1989b, 1991a), Semal and Courtois (1991), Stewart (1991b), Borovkov, Fayolle, and Korshunov (1992), Hassin and Haviv (1992), Haviv, Ritov, and Rothblum (1992), Barlow (1993a, 1993b), Meyn and Tweedie (1992, 1993a, 1993b, 1993c, 1998), O’Cinneide (1993), Lasserre (1994), Ze˘ıfman and Isaacson (1994), Ele˘ıko and Shurenkov (1996), Glynn and Meyn (1996), Borovkov (1998a), Roberts, Rosenthal, and Schwartz (1998), Tweedie (1998), Andreev, Elesin, Krylov, Kuznetsov, and Ze˘ıfman (2002), Avrachenkov and Haviv (2003), Craven (2003), Zhang and Yin (2004), Hunter (2005, 2006), and Yin George and Zhang Hanqin (2007). We would also like to refer to works related to studies of convergence for rareficated (thinned) stochastic flows. These are R´enyi (1956), Belyaev (1963), Kovalenko (1965), Mogyor´odi (1971, 1972a, 1972b), Szantai (1971a, 1971b), R˚ade (1972a, 1972b), Zakusilo (1972a, 1972b), Jagers (1974), Jagers and Lindvall (1974), Tomko (1974), Kallenberg (1975), Lindvall (1976), Serfozo (1976, 1980, 1984a, 1984b), Anisimov and Gasanenko (1978), Gasanenko (1980, 1983, 1991, 1998a, 1998b), Grandell (1980), Anisimov and Chernyak (1982, 1983), B¨oker and Serfozo (1983), Anisimov (1986, 2000), Manor (1998), and Silvestrov and Drozdenko (2006). As far as quasi-stationary distributions and related problems are concerned, the only case of non-perturbed processes was mainly studied. Models of Markov and semi-Markov type processes with discrete phase space were studied in Vere-Jones (1962, 1969), Kingman (1963), Darroch and Seneta (1965, 1967), Seneta (1966a,

8.3 Bibliographical remarks

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1966b, 1973, 1981), Seneta and Vere-Jones (1966), Daley (1968), Buiculescu (1972, 1975), Cheong (1972), Flaspohler and Holmes (1972), Pakes (1973, 1978a, 1978b, 1978/79, 1995), Flaspohler (1974), Green (1976), Pakes and Tavar´e (1981), Anisimov and Pushkin (1985), Pollett (1986, 1988a, 1990, 1991, 1995, 1999, 2001a), Abate and Whitt (1989), Pollett and Roberts (1990), Kijima (1992a, 1993a, 1997), Pollett and Vere-Jones (1992), Nair and Pollett (1993), Pollett and Stewart (1994), Ferrari, Kesten, Martinez, and Picco (1995), Jacka and Roberts (1995), van Doorn and Schrijner (1996), Ferrari, Kesten, and Martinez (1996), Hart and Pollett (1996), CoolenSchrijner, Hart, and Pollett (1999), Darlington and Pollett (2000), Elmes, Pollett, and Walker (2000), Gyllenberg and Silvestrov (1999a, 2000a), Hart and Pollett (2000), Silvestrov (2000a), Dickman and Vidigal (2002), Asselah and Castell (2003), Latouche, Remiche, and Taylor (2003), Coolen-Schrijner and van Doorn (2006a), and Ferrari, and Mari´c (2007). Quasi-stationary distributions for Markov and semi-Markov type processes with general phase space and related problems were studied in Iglehart (1974a, 1974b, 1975), Tweedie (1974a, 1974b, 1974c), Pollard and Tweedie (1975, 1976), Bolthausen (1976), Nummelin (1976, 1977, 1979, 1984), Nummelin and Tweedie (1976, 1978), Arjas and Nummelin (1977/78), Durrett (1978, 1980), Tuominen and Tweedie (1979a, 1994), Arjas, Nummelin, and Tweedie (1980/81), Pinsky (1985), Seneta and Tweedie (1985), Abadov (1984), Silvestrov and Abadov (1984, 1991, 1993), Bertoin and Doney (1994a), Ferrari, Martinez, and San Martin (1997), Klebaner, Lazar, and Zeitouni (1998), Martinez, Picco, and San Martin (1998), Breyer and Hart (2000), Lladser and San Martin (2000), Moler, Plo, and San Miguel (2000), Mei Qi Xiang and Lin Xiang (2001), Glynn and Thorisson (2001, 2002), Jacka and Warren (2002), Jacka, Lazic, and Warren (2005a, 2005b), and Kyprianou and Palmowski (2006). Chapter 4 also contains a number of lemmas. Lemmas 4.1.1–4.4.1 have a standard preparatory character. Lemmas 4.5.1–4.5.10 present a series of solidarity statements concerning existence conditions and convergence properties for moment generating functions of hitting times for perturbed semi-Markov processes. They are interesting by themselves and could have applications beyond the scope of quasi-stationary studies. Other lemmas given in Chapter 4 are of a standard character. In connection with these results, we would like to mention the works which deal with moment solidarity properties conditions of existence and upper bounds for power and exponential moments of hitting times, Kolmogorov (1937), Chung (1953, 1954, 1960), Hodges and Rosenblatt (1953), Lo`eve (1955), Pyke and Schanflie (1964), Teugels (1968b), Cheong and Teugels (1972), Kalashnikov (1973, 1978b), Silvestrov (1970, 1974, 1980a, 1983a, 1983b, 1996, 2004b), Silvestrov and Poleˇscˇ uk (1974), Poleˇscˇ uk (1977), Nummelin and Tuominen (1983), Tweedie (1983b), Gaver, Jacobs and Latouche (1984), Sbignev (1996), Pyke (1999), and Silvestrov and Drozdenko (2006). We would also like to mention some first papers on semi-Markov processes, which are Tak´acs (1954), L´evy (1954/56), Smith (1954/56), Pyke (1961a, 1961b), Janssen

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(1964, 1965), and some surveys and important books on Markov, semi-Markov and related processes, Romanovski˘ı (1949), Doob (1953), Chung Kai Lai (1960), Dynkin (1963), Gikhman and Skorokhod (1965, 1971, 1973, 1975), C ¸ inlar (1975), Kemeny, Snell, and Knapp (1966, 1976), C¸inlar (1969a), Foguel (1969), Orey (1971), Rosenblatt (1971), Korolyuk, Brodi, and Turbin (1974), Revuz (1975), Koroljuk an Turbin (1976a, 1978, 1982a), Teugels (1976, 1986), Silvestrov (1980a), Borovkov (1980, 1998a), Kovalenko, Kuznetsov, and Shurenkov (1983), Nummelin (1984), Ethier and Kurtz (1986), Anderson (1991), Korolyuk and Svishchuk (1992, 1995), Wang Zi Kun and Yang Xiang Qun (1992), Meyn and Tweedie (1993c), Kijima (1997), Ramaswami (1997), Latouche and Ramaswami (1999), Limnios, and Opris¸an (2001), Bini, Latouche, and Meini (2005), Koroliuk and Limnios (2005), and Janssen and Manca (2006, 2007), Anisimov (2008), and Harlamov (2008). The results obtained in Chapter 4 play a preparatory role and are needed for constructing asymptotic expansions for nonlinearly perturbed semi-Markov processes with finite phase space and absorption, which are given in Chapter 5. Chapter 5. This Chapter plays a key role in the book. Our main results presented in this chapter include exponential expansions in mixed ergodic and large deviation theorems for nonlinearly perturbed semi-Markov processes and the corresponding asymptotic expansions for quasi-stationary distributions. The chapter is based in part on the results obtained in the papers Gyllenberg and Silvestrov (1998, 1999a, 2000a) and Silvestrov (2000a, 2007a) and extends these results in several directions. We think that the method used for obtaining the expansions mentioned above has its own value and great potential for future studies. The first step is a standard one. It is an imbedding of the corresponding semi-Markov process with absorption to a model of regenerative process, where the regeneration times are subsequent return moments to a fixed state i ¤ 0, where 0 is the absorption state. This imbedding permits to separate the second step, connected with a construction of asymptotic expansions for power or mixed power-exponential moments of regeneration (return) times, and the third step in the corresponding algorithm, connected with a construction of asymptotic expansions for the characteristic roots ."/ . This separation significantly simplifies the asymptotic analysis. The second step is to construct asymptotic expansions for higher order moments of the hitting-without-absorption times. These moments satisfy recurrence systems of linear equations with the same perturbed coefficient matrix and the free terms connected by special recurrence systems of relations in which the free terms for the moments of a given order are given as polynomial functions of moments of lower orders. This permits to build an effective recurrence algorithm for constructing the corresponding asymptotic expansions. Each sub-step in this recurrence algorithm is a matrix but linear-type problem, where the solution of the system of linear equations with nonlinearly perturbed coefficients and free terms should be expanded in asymptotic series. The third step is to construct an asymptotic expansion for the root ."/ of the nonlin."/ ."/ ear but scalar equation 0 i i ./ D 1, where 0 i i ./ is a moment generating function

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501

for the corresponding return-without-absorption time. Here, the asymptotic expansions constructed at the second step are also essentially used. As soon as an asymptotic expansion for the characteristic root ."/ is constructed, one can use it directly to give corresponding exponential expansions in the mixed ergodic and large deviation theorems. Asymptotic expansions for quasi-stationary distributions require two more steps, which are needed for constructing asymptotic expansions at the point ."/ for the corresponding moment generation functions, giving expressions for quasi-stationary probabilities in the quotient form, and for transforming the corresponding asymptotic quotient expressions to the form of Taylor type asymptotic expansions. As a result, we get an effective algorithm for a construction of the needed asymptotic expansions. Theorems 5.4.1 and 5.4.2 give exponential expansions in mixed ergodic and large deviation theorems for nonlinearly perturbed semi-Markov processes in the pseudostationary case, while Theorems 5.4.3 and 5.4.4 give the corresponding expansions in the quasi-stationary case. Theorems 5.5.2–5.5.5 give the corresponding asymptotic expansions of quasi-stationary distributions for nonlinearly perturbed semi-Markov processes. These expansions may be compared with analogous asymptotic expansions for the usual stationary distributions given in Theorem 5.5.1. It should be noted that the construction of the asymptotic expansions for quasi-stationary distributions is much more complicated than for ordinary stationary distributions. This is so, since the quasi-stationary probabilities are functions of the characteristic roots ."/ and, therefore, are very nonlinear functionals of the corresponding moments of the transition probabilities. While ordinary stationary probabilities are simple rational functions of the corresponding stationary probabilities of imbedded Markov chains and expectations for the sojourn times. We also specify perturbation conditions in all the theorems mentioned above for an important case of nonlinearly perturbed Markov chains with absorption. The main new elements, as compared with the results listed above and given in Gyllenberg and Silvestrov (1998, 1999a, 2000a), consist in the following. Firstly, we use a more advanced and compact matrix form of the corresponding asymptotic expansions. Secondly, we also consider models with a non-empty class of non-recurrent-withoutabsorption states, which is important for applications. Thirdly, we give a detailed analysis of pivotal properties of the corresponding asymptotic expansions. Also we would like to mention Lemmas 5.1.1–5.3.9 that give asymptotic expansions for hitting probabilities, power- and mixed power-exponential moments of hitting times for nonlinearly perturbed semi-Markov processes. These results have their own values and possible applications beyond the problems studied in the book. It should be mentioned that some results similar to the results obtained in Chapter 5 were given for nonlinearly perturbed discrete time Markov chains in Silvestrov (2000a).

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A list of other works, where asymptotic expansions for some characteristics related to expectations of hitting times and stationary distributions were studied, include the papers Poliˇscˇ uk and Turbin (1973), Korolyuk and Turbin (1976a, 1978), Latouche and Louchard (1978), Louchard and Latouche (1982), Aissani and Kartashov (1983b), Abadov (1984), Kartashov (1985d, 1985e, 1996b), Latouche (1991), Hassin and Haviv (1992), Haviv, Ritov, and Rothblum (1992), Haviv and Ritov (1993), Englund and Silvestrov (1997), Cao (1998), Yin and Zhang (1998, 2003), Avrachenkov, and Lasserre (1999), Englund (1999a, 1999b, 2001), Li, Yin, G., Yin, K., and Zhang (1999), Silvestrov (2000a), Avrachenkov, Haviv, and Howlett (2001), Yin, G., Zhang Yang, and Yin, K. (2001), and Avrachenkov and Haviv (2004). Asymptotic expansions for distributions of hitting times and related functionals were studied in Korolyuk, Penev, and Turbin, (1972, 1981), Turbin (1972), Poliˇscˇ uk and Turbin (1973), Korolyuk and Turbin (1976a, 1978), Abadov (1984), Silvestrov and Abadov (1984, 1991, 1993), Korolyuk and Tadzhiev (1986), Latouche (1988), Courtois and Semal (1991), Kartashov (1985d, 1985e, 1996b), Englund and Silvestrov (1997), Englund (1999a, 1999b, 2001), Silvestrov (2000a), and Koroliuk and Limnios (2005). Papers that contains asymptotic expansions for other types of functionals for Markov type processes are Nagaev (1957, 1961), Leadbetter (1963), Gray and Pinsky (1983), Liao Ming (1988), Stewart and Sun Ji Guang (1990), Khasminskii, Yin, and Zhang (1996a, 1996b), Yin and Zhang (1998), Fuh Cheng-Der and Lai Tze Leung (2001), Gasanenko (2003), Samo˘ılenko (2006a, 2006b), and Fuh Cheng-Der (2007). It should be noted that the works listed above contain neither exponential expansions in mixed ergodic and large deviation theorems for nonlinearly perturbed semiMarkov processes, except Silvestrov and Abadov (1984, 1991, 1993), nor the corresponding asymptotic expansions for quasi-stationary distributions. We also would like to mention some works that concern with recurrence relations for different order moments for hitting time type functionals essentially used in our studies. Such recurrence relations have been obtained for Markov chains in Chung (1953, 1954, 1960). A further development was achieved in Hodges and Rosenblatt (1953), Lamperty (1960), Kemeny and Snell (1961a, 1961b), Pitman (1974, 1976, 1977), Kalashnikov (1978b), Silvestrov (1980a, 1980d, 1983b, 1996, 2004b), Nummelin and Tuominen (1983), and Tweedie (1983b). Some relevant books concerned perturbation problems for stochastic processes studied in Chapter 5 are Erd´elyi (1956), Kato (1966), Cole (1968), Korolyuk and Turbin (1976a, 1978), Wentzell and Freidlin (1979), Kevorkian and Cole (1981, 1996), Ho and Cao (1991), Kartashov (1996b), Yin and Zhang (1998), and Stewart (1998, 2001), Latouche and Ramaswami (1999), and Koroliuk and Limnios (2005). Chapters 1–5 present a theory that can be applied to study pseudo- and quasistationary phenomena in nonlinearly perturbed stochastic systems, and nonlinearly perturbed risk processes. These applications are presented, respectively, in Chapters 6 and 7.

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Chapter 6. In this chapter, we give several typical examples that show how the results of Chapters 1–5 can be applied to an analysis of pseudo- and quasi-stationary phenomena in nonlinearly perturbed stochastic systems. This chapter is partly based on the results of the papers Gyllenberg and Silvestrov (1994, 1998a, 2000a). In these papers, applications to the analysis of quasi-stationary phenomena in nonlinearly perturbed multi-server highly reliable queueing systems and metapopulation dynamic models were considered. We also consider new applications to nonlinearly perturbed M/G queueing systems with quick service and stochastic systems of birth-death type, including epidemic and population dynamics models. In all the cases, we first give conditions for mixed ergodic and limit/large deviation theorems. These results have many analogues in the prior works. Second, we formulate nonlinear perturbation conditions that permit to construct exponential expansions in mixed ergodic and limit/large deviation theorems, as well as asymptotic expansions of the corresponding quasi-stationary distributions. These results have no analogues in the preceding works, except for the papers Englund (1999a, 1999b, 2000, 2001), where analogues of some of our results were given for nonlinearly perturbed M/M/m queueing systems and Bernoulli random walks. We give references to works on limit theorems for different lifetime functionals such as occupation times or waiting times in queueing systems, lifetime in reliability models, ruin times for insurance models, etc. In this connection, we need to mention the works Kingman, (1961, 1962), Gnedenko (1964a, 1964b, 1067a, 1967b), Gnedenko and Kovalenko (1964), Solov’ev (1964, 1966, 1970, 1971, 1983), C¸inlar (1967), Pakes (1971, 1972), Masol (1972), Gnedenko and Solov’ev (1974, 1975), Kovalenko and Musaev (1974), Brown (1975, 1984), Keilson (1975, 1986), Kovalenko (1975, 1977, 1980, 1994, 1995, 2000), Zajtsev and Solov’ev (1975), Ovchinnikov (1976), Solov’ev and Sahobov (1976, 1977), Sahobov and Solov’ev (1977), Ivchenko, Kashtanov, and Kovalenko (1979), Walker (1980), Kovalenko and Kuznetsov (1981, 1988), Keilson and Sumita (1983), Anisimov and Za˘ınutdinov (1985), Butko and Korolyuk (1984), Vo˘ına and Pushkin (1984), Anisimov, Zakusilo, and Donchenko (1987), Assmussen (1987, 2003), Brown and Shao (1987), Rohlicek (1987), Kalashnikov and Rachev (1988), Anisimov and Sztrik (1989) Sztrik and Anisimov (1989), Konstantinidis and Solov’ev (1991), Solov’ev and Konstant (1991), Willekens and Teugels (1992), Kalashnikov (1994c), Zhang Hanqin, Hsu Guang-Hui, and Wang Rongxin (1995), Roy (1996), Kovalenko, Kuznetsov, Pegg (1997), Gyllenberg and Silvestrov (1998, 2000a), Kolowrocki (1998, 2000), Kovalenko, Birolini, Kuznetsov, and Shenbukher (1998), Englund (1999a, 1999b, 2001), Korolyuk (1999), Anisimov (2000, 2008), Anisimov and Kurtulush (2001), Frostig (2002), Lebedev (2002), and Korolyuk and Limnios (2005). Quasi-stationary distributions in queueing systems and related problems were studied in Pakes (1975), Thorisson (1985b), Abate and Whitt (1988, 1997, 1999), Abate, Kijima and Whitt (1991), Kijima and Makimoto (1992), Willekens and Teugels (1992), Kijima (1993b, 1995), Abate, Choudhury, and Whitt (1994a, 1994b, 1995), Choud-

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Supplements

hury and Whitt (1994), Glynn and Whitt (1994), van Doorn (1981), Gyllenberg and Silvestrov (1998, 2000a), Englund (1999a, 1999b, 2001), Gaver and Jacobs (1999) Kijima and Makimoto (1999), and Whitt (2000), while the problems of stability of stationary distributions, ergodic theorems, and some asymptotic expansions for characteristics of perturbed queueing systems were studied in Neuts and Teugels (1969), Kyprianou (1972a, 1972b), Zolotarev (1975, 1976, 1977), Kartashov (1979b, 1981, 1984, 1985a, 1985b), Tuominen and Tweedie (1979b), Borovkov (1980), Aissani and Kartashov, (1983a, 1983b), Brandt, Franken, and Lisek (1984), Franken, Kirstein, and Streller (1984), Kalashnikov and Rachev (1984, 1985, 1986, 1987), Latouche (1988), Kalashnikov and Rachev (1988), Rachev (1991), Meyn and Down (1994), Down and Meyn (1995), Asmussen and Teugels (1996), Boucherie (1997), Asmussen (1999), Anisimov and Artalejo (2002), Asmussen and O’Cinneide (2002), and Andreev, Elesin, Krylov, Kuznetsov, and Ze˘ıfman (2003). We would also like to mention some books and surveys related to asymptotic problems for queueing systems; these are Khintchine, (1955, 1963), Cox and Smith (1961), Borovkov (1972, 1980), Neuts (1981, 1989), Franken, K¨onig, Arndt, and Schmidt (1982), Kalashnikov and Rachev (1988), Kovalenko and Kuznetsov (1988), Kalashnikov (1994c), Kovalenko (1994), Kovalenko, Kuznetsov, and Pegg (1997), Limnios and Opris¸an (2001), Whitt (2002), Hassin and Haviv (2003), and Anisimov (2008). Quasi-stationary distributions and related problems for birth-and-death processes were studied in Good (1968), Solov’ev (1972), Callaert and Keilson (1973), Cavender (1978), Lindvall (1979b), Keilson and Ramaswamy (1984, 1986), Brockwell (1985), van Doorn (1987, 1991), Latouche (1987), Pakes and Pollett (1989), Ferrari, Martinez, and Picco (1991, 1992), Kijima and Seneta (1991), Kijima (1992b), Martinez (1993), Roberts and Jacka (1994), Jacka and Roberts (1995), Martinez and Vares (1995), van Doorn and Schrijner (1995a, 1995b), Bean, Bright, Latouche, Pearce, Pollett, and Taylor (1997), Kijima, Nair, Pollett, and van Doorn (1997), Roberts, Jacka, and Pollett (1997), Schrijner and van Doorn (1997), Bean, Pollett, and Taylor (1998, 2000), Cairns and Pollett (2004) van Doorn and Zeifman (2005), Coolen-Schrijner and van Doorn (2006b), van Doorn (2006), and Pollett, Zhang Hanjun, and Cairns (2007). As was mentioned above, the chapter contains an analysis of pseudo- and quasistationary phenomena for a continuous-time stochastic metapopulation model. As in other examples, the main point in our results is obtaining corresponding exponential expansions in mixed ergodic and large deviation theorems. It seems that some comments concerned the term metapopulation may be useful. Natural populations of most species have a spatial structure. Several habitat patches can support local populations. Such a set of local populations is called a metapopulation. Local populations in a metapopulation are connected by migration. A local population may go extinct due to a catastrophe. An empty patch may be colonised by migrants originating from other patches. Due to a fragmentation of natural habitats, mathematical metapopulation models have attracted an immense attention in the past few years. It is also useful to note that there is a clear analogy between metapopulation models and epidemic models

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505

and that, in fact, the epidemic models are particular cases of metapopulation models. The analogy actually takes place at different hierarchical levels. The most obvious analogy identifies patches with host individuals, colonisation with infection, and local extinction with recovery. Occupied patches correspond to infected, and empty ones to susceptible individuals. For an account of the current state of the subject as well as for many specific models and references to the literature, we refer to the works Gani (1971), Hethcote and Yorke (1984), Gyllenberg and Hanski (1992), Gyllenberg and Silvestrov (1994, 1998a, 2000a), Hanski and Gilpin (1997), Daley and Gani (1999), and Hanski (1999). In connection with our metapopulation example, we mention the papers on asymptotics of distributions of the extinction times, and quasi-stationary phenomena in population dynamic and epidemic models, as well as quasi-stationary phenomena in stochastic models of chemical reactions and other non-technical stochastic systems, which can be modelled within the framework of continuous time Markov chains or continuous time Markov processes with absorption. We refer here to the papers Weiss and Dishon (1971), Barbour (1976), Oppenheim, Shuler and Weiss (1977), Turner and Malek-Mansour (1978), Cohen (1979), Pakes (1984), Parsons and Pollett (1987), Pakes (1988a, 1988b, 1989a, 1989b,1989c, 1990), Pollett (1988b, 2001b), Kryscio and Lef`evre (1989), Durrett and Neuhauser (1991), Heyde (1981, 1884), N˚asell (1991, 1996, 1999a, 1999b, 2001, 2002a, 2002b), Al-Towaiq and Al-Eideh (1994), Gilpin and Taylor (1994), Gyllenberg and Silvestrov (1994, 1998, 2000a), Day and Possingham (1995), Kijima (1995), Hanski and Gilpin (1997), H¨ogn¨as (1997), Martin-L¨of (1998), Ramanan and Zeitouni (1999), Clancy, O’Neill, and Pollett (2001, 2003), Bobrowski (2004), Cairns and Pollett (2004), and Ross and Pollett (2006). Chapter 7. In this chapter, we present results that extend classical diffusion and Cram´er–Lundberg approximations for ruin probabilities, as well as some other ruin based functionals, to a model of nonlinearly perturbed risk processes. The chapter is based on the results obtained in the papers Gyllenberg and Silvestrov (1999b, 2000b) and Silvestrov (2000b), and Silvestrov (2007b). Our approach is based on the well-known fact that the ruin probability for classical risk process satisfies a renewal equation, if considered as a function of the initial capital. Thus, the asymptotic results for perturbed renewal equation, obtained in Chapters 1 and 2, can be applied. Using this approach we get an extension of the classical diffusion and Cram´er–Lundberg approximations for ruin probabilities to a model of nonlinearly perturbed risk processes. We introduce correction terms for the diffusion and Cram´er–Lundberg type approximations, which provide the right asymptotic behaviour of relative errors in the perturbed model. The dependence of these correction terms on the relations between the rate of the perturbation and the speed of growth of the initial capital is investigated. Various types of perturbations of risk processes are discussed. Theorems 7.2.1, 7.2.2, and 7.2.3 present new variants of the diffusion approximation, respectively, for the ruin probabilities, increments of the ruin probabilities, and

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density of the ruin probability for perturbed risk processes. These results were obtained in Silvestrov (2000b). Theorem 7.3.1 presents a new generalised unified variant of the diffusion and Cram´er-Lundberg approximations for the perturbed risk processes. Theorems 7.3.2 and 7.3.3 present exponential asymptotic expansions of the ruin probabilities for nonlinearly perturbed risk processes. These results were obtained in Gyllenberg and Silvestrov (1999b, 2000b). It is useful to note that these theorems cover, in a unified form, both the diffusion approximation case, which corresponds, in terms used in the book, to a pseudo-stationary model, as well as the Cram´er–Lundberg approximation case, which corresponds to a quasi-stationary model. In Section 7.4, we also give variants of approximations for the ruin probabilities by using the corresponding asymptotic expansions. These approximations involve higher order moments of the claim distributions and have zero asymptotic relative errors. Numerical studies show that the corresponding approximations are comparable with other approximations known in the literature. Finally, we also give the diffusion and the Cram´er– Lundberg type approximations for the distribution of the surplus prior to and at the time of the ruin for nonlinearly perturbed risk processes in Theorems 7.5.1, 7.5.2, and 7.5.3. These are new results given in Silvestrov (2007b). Our results are connected with the results related to various sufficient conditions for the diffusion and the Cram´er–Lundberg type approximations and approximations for the ruin probabilities given in Lundberg (1903, 1909, 1926), Cram´er (1926, 1930, 1955), Esscher (1932), Segerdahl (1942, 1948), Iglehart (1969), Grandell and Segerdahl (1971), Beekman (1974), Siegmund (1975), Harrison (1977), Grandell (1977, 1978/79 1991), Thorin (1977, 1982), de Vylder (1978, 1996), Gerber (1979), Janssen and Delfosse (1982), Wikstad (1983), Asmussen (1985, 1987, 1989, 2000), Glynn (1990), Sundt (1991), Schmidli (1992, 1994a, 1994b, 1995, 1997, 1999), Embrechts and Kl¨uppelberg (1993), Asmussen and Rolski (1994), Barndorff-Nielsen and Schmidli (1995), Kl¨uppelberg and Mikosch (1995), Asmussen and Kl¨uppelberg (1996), Beirlant, Teugels and Vynckier (1996), Kalashnikov and Konstantinidis (1996, 2000), de Vylder and Marceau (1996), Embrechts, Kl¨upenberg, and Mikosch (1997), Asmussen, Schmidli, and Schmidt (1999), Gyllenberg and Silvestrov (1999b, 2000b), Kalashnikov (1999), Kartashov (1999), Silvestrov (2000b, 2000c), Enikeeva, Kalashnikov, and Rusaityte (2001), Hipp and Schmidli (2004), Kartashov and Stroev (2005), and Silvestrov and Drozdenko (2006). We would also like to mention results related to asymptotics of the ruin probabilities for risk processes with heavy tailed claim distributions and more general risk type processes given in T¨acklind (1942), Andersen, E. Sparre (1957), von Bahr (1975), Thorin and Wikstad (1976), Embrechts, Goldie and Veraverbeke (1979), Embrechts and Veraverbeke (1982), Martin-L¨of (1986b), Kl¨uppelberg (1989), Embrechts and Kl¨uppelberg (1993), Asmussen, Fle, and Kl¨uppelberg (1994), Bertoin and Doney (1994b), Assmussen (1995, 1996, 2000, 2003), Kl¨uppelberg and Mikosch (1995, 1997), Asmussen and Teugels (1996), Grandell (1997), Kl¨uppelberg and Stadmular (1998), Zinchenko (1999), Bening, Korolev, and Kudryavtsev (2001), Kalashnikov and Nor-

8.3 Bibliographical remarks

507

berg (2002), Kl¨uppelberg, Kyprianou, and Maller (2004), Kl¨uppelberg and Kyprianou (2006), Ming Ruixing, Modibo Diarra, and Hu Yijun (2007). The surplus prior to and at the time of the ruin and its asymptotics have been studied for non-perturbed risk processes in Gerber, Goovarerts, and Kaas (1987), Dufresne and Gerber (1988), Dickson (1992, 2005), Dickson, Dos Reis, and Waters (1995), Willmot and Lin (1998), Schmidli (1999), Rolski, Schmidli, Schmidt, and Teugels (1999), and Badescu, Breuer, Drekic, Latouche, and Stanford (2005). Some basic books on risk theory and related topics are Cram´er (1930, 1955), Gerber (1979), Bingham, Goldie, and Teugels (1989), Grandell (1991), Sundt (1991), Daykin, Pentik¨ainen, and Pesonen (1994), de Vylder (1996), Embrechts, Kl¨uppelberg, and Mikosch (1997), Rolski, Schmidli, Schmidt, and Teugels (1999), Asmussen (2000), Bening and Korolev (2002), Mikosch (2004), and Dickson (2005).

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Index

, 24 Cnm , 284 Dh;r;j , 84 Dm;q , 80 ), 32 , 380 " ! 0, vi F, R t 28 s , 24 F .C/ n , 106 F ./ n , 106 d D, 112 d

!, 45

, 380 , 380 , 59, 380, 445 a:s: !, 33 U

!, 31 o."k /, 260 L, 24

A Absorption, 172 Algorithm multi-step reduction, 263 Approximation Cram´er–Lundberg, 439, 443, 475 diffusion, 440, 443, 475 generalised Cram´er–Lundberg, 455 generalised diffusion, 455

B Buffer bounded queue, 373

C Capital initial, 439

Chain core .i; j /-, 264 Chain of states irreducible, 262 Claim, 439 Class of non-recurrent-without-absorption states, 171 of recurrent-without-absorption states, 170 Colonisation, 426 Component cyclic, 272, 297 Condition asymptotic Cram´er type, 230 balancing, 22 Cram´er type, 57 Convergence almost sure, 33 in distribution, 45 locally uniform, 31 weak, 29, 32 Convolution, 24 r-fold, 24

D Distribution initial, 168, 169 of absorption time, 176 of first hitting time, 176 of sojourn time, 170, 174 quasi-stationary, vii, 129, 162, 241, 242, 250, 336 stationary, 114, 116, 156, 193, 204, 207, 213, 322 Distribution function arithmetic, 25 generalised non-arithmetic, 200 generating renewal equation, 24 non-arithmetic, 25

578

E Element cyclic, 262, 293 Equation characteristic, 23, 57, 62, 216, 443 improper renewal, 48 perturbed renewal, 21, 29, 61 renewal, 21, 24 Expansion .h; k/-, 478 .m; n/-matrix, 479 asymptotic, vii, 94, 155, 260, 270, 275, 285, 289, 295, 305, 324, 330, 340, 345, 403, 404, 467, 478 asymptotic matrix, 479 exponential, vi, 74, 146, 313, 372, 407, 423, 437, 459, 475 pivotal, 478 Exponent Lundberg, 440, 443

F Formula Khintchine–Pollaczek, 450 Function directly Riemann integrable, 25 forcing, 24 moment generating, 57, 216 renewal, 24 test, 228 Functions comparable, 380

G Generator of Markov chain, 350

I Independent identically distributed (i.i.d.) random variables, 26, 42 Infinitesimal, 483

K Kronecker delta-function, 27

L Lemma Choquet–Deny, 35

Index Lifetime, 362, 427 Limit renewal, 40, 70 Loop of states, 263

M Markov chain, 168 continuous-time, 350 imbedded, 169 Matrix -potential, 293 potential, 261 Measure renewal, 24 Migration, 426 Model epidemic, 425 metapopulation, 426 population, 425 Modulation semi-Markov, 360 Moment mixed power-exponential, 298 power, 280

N Norm of matrix, 481 Number of renewals, 26, 42

O Order chain asymptotic, 262 chain asymptotic -, 294 global asymptotic, 264 global asymptotic -, 294 local asymptotic, 262 local asymptotic -, 294

P Parameter perturbation, 22 Patch, 426 Period transition, 107, 108

579

Index Phenomenon pseudo-stationary, vi, 104, 167 quasi-stationary, vi, 104, 167 transient, 104, 166 Population local, 426 Probability absorption, 171, 178, 275 cyclic absorption, 196 generalised absorption, 189 generalised hitting, 189 hitting, 171, 178, 275 hitting-without-absorption, 170 ruin, 439, 442, 444 stopping, 107 transition, 168, 169, 174 Process birth-and-death, 421 imbedded semi-Markov, 374, 375 Markov renewal, 169 perturbed renewal, 42 perturbed risk, 444 perturbed semi-Markov, 173 regenerative, 106 renewal, 26, 42 risk, 439, 442, 444 semi-Markov, 169 semi-Markov of birth-and-death type, 408 semi-Markov with absorption, 170 stationary renewal, 47 Property pivotal, 262, 277, 288, 293, 309, 328 solidarity, 216, 222

R Rate gross premium, 439 Regularisation, 352

S Server main, 363 Set base, 484 Solution of renewal equation, 25

Space phase, 168, 169 State absorbing, 170 non-recurrent-withoutabsorption, 171 recurrent-withoutabsorption, 170 Step of arithmetic distribution function, 25 System M/G queueing, 373 M/M queueing, 424 of -hitting type, 295 of hitting type, 270 queueing, 362, 373 queueing highly reliable, 362 queueing with quick service, 373 System of linear equations -hitting, 297 hitting, 272

T Theorem Blackwell, 31 elementary renewal, 43 ergodic, 115, 124, 211, 213 individual ergodic, 323 mixed ergodic and large deviation, 125, 134, 256, 367, 397, 414, 433 mixed ergodic and limit, 112, 118, 215, 367, 397, 414, 433 quasi-ergodic, 144, 145, 241, 250 renewal, 26, 31, 49, 68 Skorokhod representation, 33 Time absorption, 170, 174, 176 first hitting, 170, 174, 176, 189 regeneration, 106 regenerative stopping, 107 renewal, 26 return, 173, 174, 198, 225 sojourn, 170 stopping, 107 Transition instant, 173